Invariant distribution of duplicated diffusions and application to Richardson-Romberg extrapolation
aa r X i v : . [ m a t h . P R ] J u l Invariant distribution of duplicated diffusions and applicationto Richardson-Romberg extrapolation
Vincent Lemaire ∗ , Gilles Pag`es † , Fabien Panloup ‡ November 25, 2013
Abstract
With a view to numerical applications we address the following question: given anergodic Brownian diffusion with a unique invariant distribution, what are the invariantdistributions of the duplicated system consisting of two trajectories? We mainly focuson the interesting case where the two trajectories are driven by the same Brownianpath. Under this assumption, we first show that uniqueness of the invariant distri-bution (weak confluence) of the duplicated system is essentially always true in theone-dimensional case. In the multidimensional case, we begin by exhibiting explicitcounter-examples. Then, we provide a series of weak confluence criterions (of integraltype) and also of a.s. pathwise confluence , depending on the drift and diffusion coef-ficients through a non-infinitesimal Lyapunov exponent . As examples, we apply ourcriterions to some non-trivially confluent settings such as classes of gradient systemswith non-convex potentials or diffusions where the confluence is generated by the dif-fusive component. We finally establish that the weak confluence property is connectedwith an optimal transport problem.As a main application, we apply our results to the optimization of the Richardson-Romberg extrapolation for the numerical approximation of the invariant measure ofthe initial ergodic Brownian diffusion.
Keywords : Invariant measure ; Ergodic diffusion ; Two-point motion ; Lyapunov expo-nent ; Asymptotic flatness ; Confluence ; Gradient System ; Central Limit Theorem ;Euler scheme ; Richardson-Romberg extrapolation ; Hypoellipticity; Optimal transport.
AMS classification (2000) : 60G10, 60J60, 65C05, 60F05.
When one discretizes a stochastic (or not) differential equation (SDE) by an Euler schemewith step h , a classical method to reduce the discretization error is the so-called Richardson-Romberg ( RR ) extrapolation introduced in [TT90] for diffusion processes. Roughly speak-ing, the idea of this method is to introduce a second Euler scheme with step h/ ∗ Laboratoire de Probabilit´es et Mod`eles Al´eatoires, UMR 7599, UPMC, Case 188, 4 pl. Jussieu, F-75252Paris Cedex 5, France, E-mail: [email protected] † Laboratoire de Probabilit´es et Mod`eles Al´eatoires, UMR 7599, UPMC, Case 188, 4 pl. Jussieu, F-75252Paris Cedex 5, France, E-mail: [email protected] ‡ Institut de Math´ematiques de Toulouse, Universit´e Paul Sabatier & INSA Toulouse, 135, av. deRangueil, F-31077 Toulouse Cedex 4, France, E-mail: [email protected] duplicated diffusion . Whenthe two solutions only differ by the starting value and are driven by the same Brownianmotion, the resulting coupled process is also known as 2 -point motion (terminology com-ing from the more general theory of stochastic flows, see [BS88, Car85, Kun90]). Beforebeing more specific as concerns this motivation, let us now define precisely what we call aduplicated diffusion.Consider the following Brownian diffusion solution to the stochastic differential equation(
SDE ) ≡ dX t = b ( X t ) dt + σ ( X t ) dW t , X = x ∈ R d , (1.1)where b : R d → R d and σ : R d → M ( d, q, R ) ( d × q matrices with real valued entries)are locally Lipschitz continuous with linear growth and W is a standard q -dimensionalBrownian motion defined on a filtered probability space (Ω , A , P , ( F t ) t ≥ ) (satisfying theusual conditions). This stochastic differential equation ( SDE ) has a unique strong solutiondenoted X x = ( X xt ) t ≥ . Let ρ ∈ M ( q, q, R ) be a square matrix with transpose ρ ∗ suchthat I q − ρ ∗ ρ is non-negative as a symmetric matrix. We consider a filtered probabilityspace, still denoted (Ω , A , P , ( F t ) t ≥ ) on which is defined a 2 q -dimensional standard ( F t )-Brownian motion denoted ( W, f W ) so that W and f W are two independent q -dimensionalstandard ( F t )-Brownian motions. Then we define W ( ρ ) a third standard q -dimensional( F t )-Brownian motions by W ( ρ ) = ρ ∗ W + p I q − ρ ∗ ρ f W , (1.2)which clearly satisfies h W i , W ( ρ ) ,j i t = ρ ij t, t ≥ DSDS ) is thendefined by(
DSDS ) ≡ ( dX t = b ( X t ) dt + σ ( X t ) dW t , X = x ∈ R d ,dX ( ρ ) t = b ( X ( ρ ) t ) dt + σ ( X ( ρ ) t ) dW ( ρ ) t , X ( ρ )0 = x ∈ R d . (1.3)Under the previous assumptions on b and σ , (1.3) has a unique (strong) solution. Thenboth ( X xt ) t ≥ and ( X x t , X ( ρ ) ,x t ) t ≥ are homogeneous Markov processes with transition(Feller) semi-groups, denoted ( P t ( x, dy )) t ≥ and (cid:0) Q ( ρ ) t (( x , x ) , ( dy , dy )) (cid:1) t ≥ respectively,and defined on test Borel functions f : R d → R and g : R d × R d → R , by P t ( f )( x ) = E f ( X xt ) and Q ( ρ ) t ( g )( x , x ) = E g ( X x t , X ( ρ ) ,x t ) . We will assume throughout the paper that the original diffusion X x has an uniqueinvariant distribution denoted ν i.e. satisfying νP t = ν for every t ∈ R + . The first part ofthe paper is devoted to determining what are the invariant measures of ( Q ( ρ ) t ) t ≥ (if any)depending on the correlation matrix ρ . Thus, if ρ = 0, it is clear that ν ⊗ ν is invariant2or Q (0) and if ρ = I q so is ν ∆ = ν ◦ ( x ( x, x )) − , but are they the only ones? To bemore precise, we want to establish easily verifiable criterions on b and σ which ensure that ν ∆ is the unique invariant distribution of ( DSDS ). In the sequel, we will denote by µ ageneric invariant measure of Q ( ρ ) . Now, we present the problem in more details (includingreferences to the literature). ✄ Existence of an invariant distribution for ( Q ( ρ ) t ) t ≥ . First, the family of probabilitymeasures ( µ ( ρ ) t ) t> defined on ( R d × R d , B or ( R d ) ⊗ ) by µ ( ρ ) t = 1 t Z t ν ⊗ ( dx , dx ) Q ( ρ ) s (( x , x ) , ( dy , dy )) ds (1.4)is tight since both its marginals on R d are equal to ν . Furthermore, the semi-group( Q ( ρ ) t ) t ≥ being Feller, one easily shows that any of its limiting distributions µ ( ρ ) as t → ∞ is an invariant distribution for ( Q ( ρ ) t ) t ≥ such that µ ( ρ ) ( dx × R d ) = µ ( ρ ) ( R d × dx ) = ν ( dx ).Also note that, if uniqueness fails and ( P t ) t ≥ has two distinct invariant distributions ν and ν ′ , a straightforward adaptation of the above (sketch of) proof shows that ( Q ( ρ ) t ) t ≥ has (at least) an invariant distribution with marginals ( ν, ν ′ ) and another with ( ν ′ , ν ) asmarginals. ✄ Uniqueness of the invariant distribution of ( Q ( ρ ) t ) t ≥ . It is clear that in full generalitythe couple (
X, X ( ρ ) ) may admit several invariant distributions even if X has only one suchdistribution. So is the case when σ ≡ x, t ) of the ODE ≡ ˙ x = b ( x ) has 0as a unique repulsive equilibrium and a unique invariant distribution ν on R d \ { } . Thenboth distributions ν ⊗ and ν ∆ (defined as above) on ( R d \ { } ) are invariant and if ν isnot reduced to a Dirac mass (think e.g. to a 2-dimensional ODE with a limit cycle around0) ( DSDS ) has at least two invariant distributions.In the case ( σ
0) the situation is more involved and depends on the correlationstructure ρ between the two Brownian motions W and W ( ρ ) . The diffusion matrixΣ( X x t , X ( ρ ) ,x t ) of the couple ( X x , X ( ρ ) ,x ) at time t > ξ , ξ )Σ( ξ , ξ ) ∗ = (cid:20) σσ ∗ ( ξ ) σ ( ξ ) ρσ ∗ ( ξ ) σ ( ξ ) ρ ∗ σ ∗ ( ξ ) σσ ∗ ( ξ ) (cid:21) ( e.g. the square root in the symmetric non-negative matrices or the Choleski transform. . . ).First, note that if I q − ρ ∗ ρ is positive definite as a symmetric matrix, it is straight-forward that ellipticity or uniform ellipticity of σσ ∗ (when q ≥ d ) for X x is transferredto Σ( X x t , X ( ρ ) ,x t )Σ( X x t , X ( ρ ) ,x t ) ∗ for the couple ( X x , X ( ρ ) ,x ). Now, uniform ellipticity,combined with standard regularity and growth/boundedness assumption on the coeffi-cients b , σ and their partial derivatives, classically implies the existence for every t > p t ( x, y ) for X xt . These additional conditionsare automatically satisfied by the “duplicated coefficients” of ( DSDS ). At this stage,it is classical background that any homogeneous Markov process whose transition has a(strictly) positive density for every t > b and σ which ensure uniqueness ofthe invariant distribution ν for X , we get uniqueness for the “duplicated” diffusion process( X, X ( ρ ) ) as well.The hypo-elliptic case also implies the existence of a density for X xt and the uniquenessof the invariant distribution under controllability assumptions on a companion differen-tial system of the SDE. This property can also be transferred to ( DSDS ), although the3roof becomes significantly less straightforward than above (see Appendix B for a precisestatement and a detailed proof).We now consider one of the main problems of this paper: the degenerate case ρ = I q .This corresponds to W ( ρ ) = W so that X ( ρ ) ,x = X x , i.e. ( DSDS ) is the equation of the2-point motion in the sense of [Kun90] section 4.2 and [Har81]. This 2-point motion hasbeen extensively investigated (see [Car85]) from an ergodic viewpoint, especially when theunderlying diffusion, or more generally the stochastic flow Φ( ω, x, t ) lives on a (smooth)compact Riemannian manifold M . When this flow is smooth enough in x , the long runbehaviour of such a flow (under its steady regime) can be classified owing to its Lyapunovspectrum. For what we are concerned with, this classification is based on the top Lyapunovexponent defined by λ := lim sup t → + ∞ t log k D x Φ( x, t ) k where k D x Φ( x, t ) k denotes the operator norm of the differential (tangent) of the flow.In this compact setting and when the top Lyapunov exponent is positive, the long runbehaviour of the two-point on M \ ∆ has been deeply investigated in [BS88] (see also[DKK04] for further results in this direction). Such an assumption implies that ∆ issomewhat repulsive.Here, we are in fact concerned with the opposite case. Our aim is to identify naturalassumptions under which the invariant distribution of the 2-point motion is unique (henceequal to ν ∆ ). It seems clear that these conditions should in some sense imply that thepaths cluster asymptotically either in a pathwise or in a statistical sense. When λ < b and σ of the original R d -valued diffusion, not assumed to be smooth. Moreprecisely, we show in Section 2 that in the one-dimensional case, uniqueness of ν ∆ is almostalways true (as soon as ( SDE ) has a unique invariant distribution) and that under someslightly more constraining conditions, the diffusion is pathwise confluent ( i.e. pathwiseasymptotic clustering holds). This second result slightly extends by a different method aresult by Has’minskii in [Has80].Section 3 is devoted to the multidimensional framework. We first provide a simple counter-example where uniqueness of ν ∆ does not hold. Then, we obtain some sharp criterionsfor uniqueness. We begin by a general uniqueness result (for ν ∆ ) (Theorem 3.2) involvingin an Euclidean framework (induced by a positive definite matrix S and its norm | . | S ) a pseudo-scale function f θ designed from a non-negative continuous function θ : R + → R .Basically both uniqueness and pathwise confluence follow fromconditions involving the coefficients of the diffusion b and σ , S and θ , combined with arequested behavior of the pseudo-scale function at 0 + . The main ingredient of the proof isBirkhoff’s ergodic Theorem applied to the one-dimensional Itˆo process f θ ( | X x t − X x t | S ).Using additional martingale arguments, we also establish that the asymptotic pathwiseconfluence holds under slightly more stringent conditions.Then, in Subsection 3.3, we draw a series of corollaries of Theorem 3.2 (illustrated on fewexamples) which highlight easily verifiable conditions. To this end we introduce a functionΛ S : R d × R d \ ∆ R d × R d → R called Non-Infinitesimal S -Lyapunov (NILS) exponent defined4or every x, y ∈ R d , x = y byΛ S ( x, y ) = ( b ( x ) − b ( y ) | x − y ) S | x − y | S + 12 k σ ( x ) − σ ( y ) k S | x − y | S − (cid:12)(cid:12)(cid:12) ( σ ∗ ( x ) − σ ∗ ( y )) S ( x − y ) | x − y | S (cid:12)(cid:12)(cid:12) ! . (1.5)In particular we show (see Corollary 3.2) that if, for every probability measure m on c ∆ R d × R d such that m ( dx × R d ) = m ( R d × dy ) = ν , Z R d × R d Λ S ( x, y ) m ( dx, dy ) < ν ∆ is unique and if furthermore Λ S ≤ − c < R d × R d , then pathwise confluence holds true. Moreover, under a directionalellipticity condition on σ , we show that the negativity of Λ S (at least in an integratedsense) can be localized near the diagonal (see Subsection 3.3 for details). A differentialversion of the criterion is established when b and σ are smooth (see Corollary 3.3).Note that these criterions obtained in the case ρ = I q can be extended to the (last) case ρ ∗ ρ = I q using that W ( ρ ) = ρW is still a standard B.M. (think to ρ = − d = 1).For the sake of simplicity (and since it is of little interest for the practical implementationof the Richardson-Romberg extrapolation), we will not consider this case in the paper.Then, we give some examples and provide an application to gradient systems ( b = −∇ U and a constant σ function). In particular, we obtain that our criterions can be appliedto some situations where the potential is not convex. More precisely, we prove that fora large class of non-convex potentials, super-quadratic at infinity , the 2-point motion isweakly confluent if the diffusive component σ is sufficiently large. Furthermore, in theparticular case U ( x ) = ( | x | − , we prove that the result is true for every σ > Integrated
NILS, the weak confluence property can be connected with an optimal transport problem.The second part of the paper (Section 4) is devoted to a first attempt in a long runergodic setting to combine the Richardson-Romberg extrapolation with a control of thevariance of this procedure (see [Pag07] in a finite horizon framework). To this end weconsider two Euler schemes with decreasing steps γ n and ˜ γ n satisfying ˜ γ n − = ˜ γ n = γ n / ρ -correlated Brownian motion increments. We show that the optimal efficiency of theRichardson-Romberg extrapolation in this framework is obtained when ρ = I q , at leastwhen the above uniqueness problem for ν ∆ is satisfied. To support this claim we establisha Central Limit Theorem whose variance is analyzed as a function of ρ . Notations. • | x | = √ xx ∗ denotes the canonical Euclidean norm of x ∈ R d ( x ∗ transposeof the column vector x ). • k A k = p Tr( AA ∗ ) if A ∈ M ( d, q, R ) and A ∗ is the transpose of A (which is but thecanonical Euclidean norm on R d ). • ∆ R d × R d = { ( x, x ) , x ∈ R d } denotes the diagonal of R d × R d . • S ( d, R ) = { S ∈ M ( d, d, R ) , S ∗ = S } , S + ( d, R ) the subset of S ( d, R ) of non-negativematrices, S ++ ( d, R ) denotes the subset of positive definite such matrices and √ S denotesthe unique square root of S ∈ S + ( d, R ) in S ++ ( d, R ) (which commutes with S ). x ⊗ y = xy ∗ = [ x i y j ] ∈ M ( d, d, R ), x, y ∈ R d . • If S ∈ S ++ ( d, R ), we denote by ( . | . ) S and by | . | S , the induced inner product andnorm on R d , defined by ( x | y ) S = ( x | Sy ) and | x | S = ( x | x ) S respectively. Finally, for A ∈ M ( d, d, R ), we set k A k S = Tr( A ∗ SA ). 5 µ n ( R d ) = ⇒ µ denotes the weak convergence of the sequence ( µ n ) n ≥ of probability measuresdefined on ( R d , B or ( R d )) toward the probability measure µ . P ( X, A ) denotes the set ofprobability distributions on ( X, A ). • For every function f : R d → R , define the Lipschitz coefficient of f by [ f ] Lip =sup x = y | f ( x ) − f ( y ) || x − y | ≤ + ∞ . We first show that, in the one-dimensional case d = q = 1, uniqueness of ν implies that ν ∆ , as defined in the introduction, is the unique invariant distribution of the duplicateddiffusion. The main theorem of this section is Theorem 2.1 which consists of two claims.The first one establishes this uniqueness claim using some ergodic-type arguments. Notethat we do not require that σ never vanishes. The second claim is an asymptotic pathwiseconfluence property for the diffusion in its own scale, established under some slightly morestringent assumptions involving the scale function p , see below. This second result, underslightly less general assumptions, is originally due to Has’minskii (see [Has80], Appendixto the English edition, Theorem 2.2, p.308). It is revisited here by different techniques,mainly comparison results for one dimensional diffusions and ergodic arguments. Notethat uniqueness of ν ∆ can always be retrieved from asymptotic confluence (see Remark2.1).Before stating the result, let us recall some definitions. We denote by M the speedmeasure of the diffusion classically defined by M ( dξ ) = ( σ p ′ ) − ( ξ ) dξ , where p is the scalefunction defined (up to a constant) by p ( x ) = Z xx dξe − R ξx bσ ( u ) du , x ∈ R . Obviously, we will consider p only when it makes sense as a finite function (so is the caseif b/σ is locally integrable on the real line). We are now in position to state the result. T HEOREM
Assume that b and σ are continuous functions on R being such that strongexistence, pathwise uniqueness and the Feller Markov property hold for ( SDE ) from any x ∈ R . Assume furthermore that there exists λ : R + → R + , strictly increasing, with λ (0) = 0 and R + λ ( u ) − du = + ∞ such that for all x, y ∈ R , | σ ( y ) − σ ( x ) | ≤ λ ( | x − y | ) .Then, the following claims hold true. ( a ) If ( X t ) t ≥ admits a unique invariant distribution ν , the distribution ν ∆ = ν ◦ ( ξ ( ξ, ξ )) − is the unique invariant measure of the duplicated diffusion ( X x t , X x t ) t ≥ . ( b ) (Has’minskii) Assume that the scale function p is well-defined as a real function onthe real line and that, lim x →±∞ p ( x ) = ±∞ and M is finite . Then, ν = M/M ( R ) is the unique invariant distribution of ( X t ) t ≥ and ( p ( X t )) t ≥ ispathwise confluent: P - a.s. , for every x , x ∈ R , p ( X x t ) − p ( X x t ) tends to 0 when t → + ∞ . R EMARK . ✄ The general assumptions on b and σ are obviously fulfilled wheneverthese functions are locally Lipschitz with linear growth.6 The proofs of both claims are based on (typically one-dimensional) comparison argu-ments. This also explains the assumption on σ which is a classical sufficient assumptionto ensure comparison of solutions, namely, if x ≤ x , then X x t ≤ X x t for every t ≥ a.s. (see [IW77]). ✄ The additional assumptions made in ( b ) imply the uniqueness of ν (see the proof below).The uniqueness of the invariant distribution ν ∆ for the duplicated diffusion follows by( a ). However, it can also be viewed as a direct consequence of the asymptotic pathwiseconfluence of p ( X x i t ), i = 1 , t → + ∞ . Actually, if for all x , x ∈ R d , p ( X x t ) − p ( X x t ) t → + ∞ −−−−→ a.s , we deduce that for any invariant distribution µ of ( X x , X x ) andevery K > Z R (cid:16) | p ( x ) − p ( x ) | ∧ K (cid:17) µ ( dx , dx ) ≤ lim sup t → + ∞ t Z t E µ (cid:16) | p ( X x s ) − p ( X x s ) | ∧ K (cid:17) ds = 0 . As a consequence, p ( x ) = p ( x ) µ ( dx , dx )- a.s. Since p is an increasing function, itfollows that µ ( { ( x, x ) , x ∈ R } ) = 1 and thus that µ = ν ∆ . ✄ As mentioned before, ( b ) slightly extends a result by Has’minskii obtained in [Has80]with different methods and under the additional assumption that σ never vanishes (whereaswe only need the scale function p to be finite which allows e.g. for the existence of in-tegrable singularities of bσ ). Note however that the case of an infinite speed measure M (which corresponds to null recurrent diffusions) is also investigated in [Has80], requiringextra non-periodicity assumptions on σ . Proof. ( a ) Throughout the proof we denote by ( X x t , X x t ) the duplicated diffusion at time t ≥ Q t (( x , x ) , dy , dy )) t ≥ its Feller Markov semi-group. The set I DSDS ofinvariant distributions of ( Q t ) t ≥ is clearly nonempty, convex and weakly closed. Sinceany such distribution µ has ν as marginals (in the sense µ ( dx × R ) = µ ( R × dx ) = ν ), theset I DSDS is tight and consequently weakly compact in the the topological vector spaceof signed measures on ( R , B or ( R )) equipped with the weak topology. As a consequenceof the Krein-Millman Theorem, I DSDS admits extremal distributions and is the convexhull of these extremal distributions.Let µ be such an extremal distribution and consider the following three subsets of R : A + = { ( x , x ) , x > x } , A − = { ( x , x ) , x > x } and A = { ( x, x ) , x ∈ R } = ∆ R . We first want to show that if µ ( A + ) > µ A + defined by µ A + = µ ( . ∩ A + ) µ ( A + ) is also an invariant distribution for ( Q t ) t ≥ . Under the above assumptionson b and σ , one derives from classical comparison theorems and strong pathwise uniquenessarguments for the solutions of ( SDE ) (see e.g. [IW77]) that ∀ ( x , x ) ∈ c A + = R \ A + , Q t (( x , x ) , c A + ) = 1 . We deduce that for every ( x , x ) ∈ R and t ≥ Q t (( x , x ) , A + ) = P (cid:0) ( X x t , X x t ) ∈ A + (cid:1) = A + ( x , x ) P ( τ x ,x > t )where τ x ,x = inf { t ≥ , X x t ≤ X x t } . The second equality follows from the pathwiseuniqueness since no bifurcation can occur. Now, let µ ∈ I DSDS . Integrating the aboveequality and letting t go to infinity implies µ ( A + ) = Z A + µ ( dx , dx ) P ( τ x ,x = + ∞ ) . µ ( A + ) >
0, then µ ( dx , x )- a.s. P ( τ x ,x = + ∞ ) = 1 on A + i.e. X x t > X x t for every t ≥ a.s. . As a consequence, µ ( dx , dx )- a.s. , for every B ∈ B ( R d × R d ), ( x ,x ) ∈ A + Q t (( x , x ) , B ) = ( x ,x ) ∈ A + Q t (( x , x ) , B ∩ A + ) = Q t (( x , x ) , B ∩ A + )where we used again that Q t (( x , x ) , A + ) = 0 if x ≤ x . Then, since µ is invariant, wededuce from an integration of the above equality that µ ( B ∩ A + ) = Z R Q t (( x , x ) , B ) ( x ,x ) ∈ A + µ ( dx , dx ) . It follows that if µ ( A + ) > µ A + is invariant.If µ ( A + ) <
1, one shows likewise that µ c A + an invariant distribution for ( Q t ) t ≥ as well.Then, if µ ( A + ) ∈ (0 , µ is a convex combination of elements of I DSDS µ = µ ( A + ) µ A + + µ ( c A + ) µ c A + so that µ cannot be extremal. Finally µ ( A + ) = 0 or 1.Assume µ ( A + ) = 1 so that µ = µ ( . ∩ A + ). This implies that X > X P µ - a.s. . But µ being invariant, both its marginals are ν i.e. X and X are ν -distributed. This yieldsa contradiction. Indeed, let ϕ be a bounded increasing positive function. For instance,set ϕ ( u ) := 1 + u √ u +1 , u ∈ R . Then, E [ ϕ ( X ) − ϕ ( X )] > X > X P µ - a.s. butwe also have E [ ϕ ( X ) − ϕ ( X )] = 0 since X and X have the same distribution. Thiscontradiction implies that µ ( A + ) = 0.One shows likewise that µ ( A − ) = 0 if µ is an extremal measure. Finally any extremaldistribution of I DSDS is supported by A = ∆ R . Given the fact that the marginalsof µ are ν this implies that µ = ν ∆ = ν ◦ ( x ( x, x )) − which in turn implies that I DSDS = { ν ∆ } .( b ) Since the speed measure M is finite and σ never vanishes, the distribution ν ( dξ ) = M ( dξ ) /M ( R ) is the unique invariant measure of the diffusion. Thus, by ( a ), we also havethe uniqueness of the invariant distribution for the duplicated diffusion.Let x , x ∈ R . If x > x then X x t ≥ X x t , still by a comparison argument, and p ( X x t ) ≥ p ( X x t ) since p isincreasing. Consequently M x ,x t = p ( X x t ) − p ( X x t ), t ≥
0, is a non-negative continuouslocal martingale, hence P - a.s. converging toward a finite random limit ℓ x ,x ∞ ≥
0. Oneproceeds likewise when x < x (with ℓ x ,x ∞ ≤ x = x , M t = ℓ x ,x ∞ ≡
0. Theaim is now to show that ℓ x ,x ∞ = 0 a.s. To this end, we introduce µ t ( dy , dy ) := 1 t Z t Q s (( x , x ) , dy , dy ) ds, ( x , x ) ∈ R d × R d and we want to check that for every ( x , x ) ∈ R d × R d , ( µ t ( dy , dy )) t ≥ converges weaklyto ν ∆ . Owing to the uniqueness of ν ∆ established in ( a ) and to the fact that any weaklimiting distribution of ( µ t ( dy , dy )) t ≥ is always invariant (by construction), it is enoughto prove that ( µ t ( dy , dy )) t ≥ is tight. Since the tightness of a sequence of probabilitymeasures defined on a product space is clearly equivalent to that of its first and secondmarginals, it is here enough to prove the tightness of ( t − R t P s ( x , dy ) ds ) t ≥ for any x ∈ R .Let x ∈ R . Owing to the comparison theorems, we have for all t ≥ M ∈ R ,8 t ( x , [ M, + ∞ )) ≤ P t ( x, [ M, + ∞ )) if x ≥ x and P t ( x , ( −∞ , M ]) ≤ P t ( x, ( −∞ , M ]) if x ≥ x . Since ν is invariant and equivalent to the Lebesgue measure, we deduce that P t ( x , [ M, + ∞ )) ≤ ν ([ M, + ∞ )) ν ([ x , + ∞ )) and P t ( x , ( −∞ , M )) ≤ ν (( −∞ , M )) ν (( −∞ , x ]) . The tightness of ( P t ( x , dy )) t ≥ follows (from that of ν ) and we derive from what preceedsthat ∀ ( x , x ) ∈ R d × R d , t Z t Q s (( x , x ) , dy , dy ) ds ( R d ) = ⇒ ν ∆ ( dy , dy ) . Now, note that for every L ∈ N , the function g L : ( y , y )
7→ | p ( y ) − p ( y ) |∧ L is continuousand bounded. Hence by C´esaro’s Theorem, we have that1 t Z t Q s ( g L )( x , x ) ds = 1 t Z t E g L ( X x s , X x ) ds −→ E ( | ℓ x ,x ∞ | ∧ L )whereas, by the above weak convergence of ( µ t ( dy , dy )) t ≥ , we get1 t Z t Q s ( g L )( x , x ) ds −→ Z R d g L ( y , y ) ν ∆ ( dy , dy ) = 0 as t → + ∞ since g L is identically 0 on ∆ R d × R d . It follows, by letting L go to infinity, that E | ℓ x ,x ∞ | = 0 . This implies ℓ x ,x ∞ = 0 P - a.s. which in turn implies that P - a.s. p ( X x t ) − p ( X x t ) −→ t → + ∞ . Finally, it remains to prove that we can exchange the quantifiers, i.e. that P - a.s. , p ( X x t ) − p ( X x t ) −→ x , x . Assume that x ≥ x . Again by the comparison theoremand the fact that p increases, we have 0 ≤ p ( X x t ) − p ( X x t ) ≤ p ( X ⌊ x ⌋ +1 t ) − p ( X ⌊ x ⌋ t ). Thismeans that we can come down to a countable set of starting points. ✷ In the continuity of the second part of Theorem 2.1( b ), it is natural to wonder whether aone-dimensional diffusion is asymptotically confluent, i.e. when for all x , x ∈ R , X x t − X x t tends to 0 a.s as t → + ∞ . In the following corollary, we show that such propertyholds in a quite general setting. C OROLLARY ( a ) Assume the hypothesis of Theorem 2.1 ( b ) hold. If furthermore, σ never vanishes and lim sup | x |→ + ∞ Z x bσ ( ξ ) dξ < + ∞ then, P - a.s. , for every x , x ∈ R , X x t − X x t −→ as t → + ∞ . ( b ) The above condition is in particular satisfied if there exists
M > such that | x | > M = ⇒ sign( x ) b ( x ) ≤ . roof. ( a ) Under the assumptions of the theorem, p is continuously differentiable on R and p ′ ( x ) = e − R xx bσ ( u ) du , x ∈ R . Then it is clear that p ′ inf = inf x ∈ R p ′ ( x ) > | x |→ + ∞ Z xx bσ ( ξ ) dξ < + ∞ . By the fundamentaltheorem of calculus, we know that, | X x t − X x t | ≤ p ′ min | p ( X x t ) − p ( X x t ) | and the result follows from Theorem 2.1( b ).( b ) Since σ never vanishes, p ′′ is well-defined and for every x ∈ R , p ′′ ( x ) = − b ( x ) p ′ ( x ) σ ( x ) .Using that p ′ is positive, we deduce from the assumptions that ∃ M > ( p ′′ ( x ) ≥ , x ≥ Mp ′′ ( x ) ≤ , x ≤ − M. Now, p ′ being continuous, it follows that p ′ attains a positive minimum p ′ min > Examples. 1.
Let U be a positive a twice differentiable function such that lim | x |→ + ∞ U ( x ) =+ ∞ and consider the one-dimensional Kolmogorov equation dX t = − U ′ ( X t ) dt + σdW t with σ >
0. Then, lim inf | x |→ + ∞ xU ′ ( x ) > σ ⇒ X xt − X yt t → + ∞ −−−−→ a.s. Note that in particular, this result holds true even if U has several local minimas. Let σ : R → (0 , + ∞ ) be a locally Lipschitz continuous function with linear growth sothat the SDE dX t = σ ( X t ) dW t defines a (Markov) flow ( X xt ) t ≥ of local martingales. If σ ∈ L ( R , B or ( R ) , λ ) then thereexists a unique invariant measure ν ( dξ ) = c σ dξσ ( ξ ) and ( X x i t ) t ≥ , i = 1 , b )) since p ( x ) = x . Note that the linear growthassumption cannot be significantly relaxed since a stationary process cannot be a truemartingale which in turn implies that ν has no (finite) first moment. In this section, we begin by an example of a multidimensional Brownian diffusion ( X x , X x )for which ν ∆ (image of ν on the diagonal) is not the only one invariant distribution. Thus,Theorem 2.1 is specific to the case d = 1 and we can not hope to get a similar result forthe general case d ≥
2. It is of course closely related to the classification of two-pointmotion on smooth compact Riemannian manifolds since the unit circle will turn out to bea uniform attractor of the diffusion. 10 .1 Counterexample in -dimension Roughly speaking, saying that ν ∆ is the only one invariant distribution means in a sensethat X xt − X yt has a tendency to converge towards 0 when t → + ∞ . Thus, the idea in thecounterexample below is to build a “turning” two-dimensional ergodic process where theangular difference between the two coordinates does not depend on t . Such a constructionleads to a model where the distance between the two coordinates can not tend to 0 (Notethat some proofs are deferred to Appendix B). We consider the 2-dimensional SDE withLipschitz continuous coefficients defined ∀ x ∈ R by b ( x ) = (cid:16) x { ≤| x |≤ } − x | x | {| x |≥ } (cid:17) (1 − | x | ) σ ( x ) = ϑ Diag( b ( x )) + (cid:20) − cx cx , (cid:21) where ϑ, c ∈ (0 , + ∞ ) are fixed parameters.Switching to polar coordinates X t = ( r t cos ϕ t , r t sin ϕ t ), t ∈ R + , we obtain that this SDE also reads dr t = min( r t , − r t )( dt + ϑdW t ) , r ∈ R + (3.6) dϕ t = c dW t , ϕ ∈ [0 , π ) , (3.7)where x = r (cos ϕ , sin ϕ ) and W = ( W , W ) is a standard 2-dimensional Brownianmotion.Standard considerations about Feller classification (see Appendix B for details) show that,if x = 0 ( i.e. r >
0) and ϑ ∈ (0 , √
2) then r t −→ t → + ∞ , (3.8)while it is classical background that P - a.s. ∀ ϕ ∈ R + , t Z t δ e i ( ϕ cW s ) ds = ⇒ λ S as t → + ∞ where S denotes the unit circle of R . Combining these two results straightforwardlyyields ∀ x ∈ R \ { (0 , } , P - a.s. t Z t δ X xs ds ( R ) = ⇒ λ S as t → + ∞ . On the other hand, given the form of ϕ t , it is clear that if x = r e iϕ and x ′ = r ′ e iϕ ′ , r , r ′ = 0, ϕ , ϕ ′ ∈ [0 , π ), thenlim t → + ∞ | X xt − X x ′ t | = | e i ( ϕ − ϕ ′ ) − | which in turn implies thatlim t → + ∞ t Z t | X xs − X x ′ s | ds = | e i ( ϕ − ϕ ′ ) − | P - a.s. This limit being different from 0 as soon as ϕ = ϕ ′ , one derives, as a consequence, that ν ∆ cannot be the only invariant distribution. In fact, a more precise statement can beproved. 11 ROPOSITION ( a ) A distribution µ is invariant for the semi-group ( Q t ) t ≥ of theduplicated diffusion if and only if µ has the following form: µ = L ( e i Θ , e i (Θ+ V ) ) (3.9) where Θ is uniformly distributed over [0 , π ] and V is a [0 , π ) -valued random variableindependent of Θ( b ) When V = 0 a.s. , we retrieve ν ∆ whereas, when V also has uniform distribution on [0 , π ] , we obtain ν ⊗ ν . Finally, µ is extremal in the convex set of ( Q t ) t ≥ invariantdistributions if and only if there exists θ ∈ [0 , π ) such that V = θ a.s . The proof is postponed to Appendix B. However, note that the claim about extremalinvariant distributions follows from the fact that for every θ ∈ [0 , π ), ( Q t ) t ≥ leaves theset Γ θ := { ( e iϕ , e iϕ ′ ) ∈ S × S , ϕ ′ − ϕ ≡ θ mod . π } stable. R EMARK . In the above counterexample, the invariant measure of ( r t ) t ≥ is the Diracmeasure δ . In fact, setting again x = ε e iϕ and x ′ = r ′ e iϕ ′ and using that X xt − X x ′ t = r xt (cid:16) e i ( ϕ + W t ) − e i ( ϕ ′ + W t ) (cid:17) + ( r xt − r x ′ t ) e i ( ϕ ′ + W t ) , an easy adaptation of the above proofshows that it can be generalized to any ergodic non-negative process ( r t ) t ≥ solution toan autonomous SDE and satisfying the following properties: • Its unique invariant distribution π satisfies π ( R ∗ + ) = 1. • For every x, y ∈ (0 , + ∞ ), r xt − r yt −→ a.s. as t → + ∞ .For instance, let ( X xt ) t ≥ be an Ornstein-Uhlenbeck process satisfying the SDE dX t = − X t dt + σdW t , X = x . Set r xt = ( X xt ) (this is a special case of the Cox-Ingersoll-Rossprocess). The process ( r xt ) clearly satisfies the first two properties. Furthermore, ( X xt ) t ≥ satisfies a.s. for every x, y ∈ R and every t ≥ | X xt − X yt | = | x − y | e − t . Then, since forevery x ∈ R , X xt t = − t Z t X xs ds + σ W t t → a.s. as t → + ∞ , it follows that ( r xt ) t ≥ also satisfies for all positive x, y , r xt − r yt −→ a.s. as t → + ∞ (Many other examples can be built using Corollary 2.1).Finally, note that if µ = L ( Re i Θ , Re i (Θ+ V ) ) where R , Θ and V are independent randomvariables such that the distributions of R and Θ are respectively π and the uniform dis-tribution on [0 , π ] and V takes values in [0 , π ), then µ is an invariant distribution of theassociated duplication system.In connection with this counterexample we can mention a general result on the Brow-nian flows of Harris (see [Har81],[Kun90] Theorem 4.3.2). The theorem gives conditionson b and σ under which ν is an invariant measure of the one point motion ( X xt ) t ≥ and ν ⊗ ν is an invariant measure of the two point motion ( X x t , X x t ) t ≥ . ( S, θ ) -confluence In the sequel of this section, we propose criterions for the uniqueness of the invariantdistribution of the duplicated system in the multidimensional case. The underlying ideaof the criterions discussed below is to analyze the coupled diffusion process ( X x , X x )through the squared distance process r t = | X x t − X x t | S (where we recall that for a givenpositive definite matrix S , | . | S is the Euclidean norm on R d induced by the scalar product12 x | y ) S = ( x | Sy )). It is somewhat similar to that of Has’minskii’s test for explosion ofdiffusions in R d or to the one proposed in Chen and Li’s work devoted to the couplingof diffusions (see [CL89]). We begin by a general abstract result under an assumptiondepending on a continuous function θ : (0 , + ∞ ) → R + to be specified further on. Then,more explicit pointwise or integrated criterions are derived in the next subsections. Inparticular, one involves a kind of bi-variate non-infinitesimal Lyapunov exponent.Let us introduce some notations. For some probability measures ν and ν ′ on R d , weset P ⋆ν,ν ′ = n m ∈ P ( R d × R d ) , m ( dx × R d ) = ν, m ( R d × dy ) = ν ′ , m (∆ R d × R d ) = 0 o and P ⋆ = {P ⋆ν,ν ′ , ν, ν ′ ∈ I SDE } where I SDE denotes the set of invariant distributions of(
SDE ). In particular, P ⋆ = P ⋆ν,ν when I SDE = { ν } (which is the case of main interest).For S ∈ S ++ ( d, R ), we also set[ b ] S, + = sup x = y ( b ( x ) − b ( y ) | x − y ) S | x − y | S . Note that if [ b ] S, + < + ∞ and if σ is Lipschitz continuous, strong existence, pathwiseuniqueness and the Feller Markov property hold for ( SDE ).For a continuous function θ : (0 , + ∞ ) → R + , we define the pseudo-scale C -function f θ and its companion g θ by ∀ u ∈ (0 , + ∞ ) , f θ ( u ) = Z u e R ξ θ ( w ) w dw dξ and g θ ( u ) = uf ′ θ ( u ) . (3.10)Finally, for S and θ defined as above, we define the ( S, θ ) -confluence function Ψ θ,S on c ∆ R d byΨ θ,S ( x, y ) = ( b ( x ) − b ( y ) | x − y ) S + 12 k σ ( x ) − σ ( y ) k S − θ ( | x − y | S ) (cid:12)(cid:12)(cid:12) ( σ ∗ ( x ) − σ ∗ ( y )) S ( x − y ) | x − y | S (cid:12)(cid:12)(cid:12) . Let us now state the result. T HEOREM
Let S ∈ S ++ ( d, R ) . Assume that b is a continuous function such that [ b ] S, + < + ∞ and σ is Lipschitz continuous. Assume that the set I SDE of invariant dis-tributions of
SDE is (nonempty, convex and) weakly compact. Furthermore, assume thatfor every m ∈ P ⋆ , the following ( S, θ ) -confluence condition is satisfied: there exists acontinuous function θ : (0 , + ∞ ) → R + such that ( i ) lim sup u → + Z u θ ( w ) − w dw < + ∞ . ( ii ) Z R d × R d f ′ θ ( | x − y | S )Ψ θ,S ( x, y ) m ( dx, dy ) < . (3.11)( a ) Weak confluence (Uniqueness of both invariant distributions) : Then, if I DSDS denotesthe set of invariant distributions of the duplicated system ( DSDS ) , one has I SDE = (cid:8) ν (cid:9) and I DSDS = (cid:8) ν ∆ (cid:9) keeping in mind that ν ∆ = ν ◦ ( x ( x, x )) − . b ) Pathwise confluence: Let θ : (0 , + ∞ ) → R + be a continuous function such that Z e R v θ ( w ) w dw dv < + ∞ and such that ∀ x, y ∈ R d , x = y, Ψ θ,S ( x, y ) < . If furthermore, for every x ∈ R d , (cid:16) t Z t P s ( x, dy ) ds (cid:17) t ≥ is tight, we have a.s. pathwiseasymptotic confluence: ∀ x , x ∈ R d , X x t − X x t −→ as t → + ∞ P - a.s. (3.12) R EMARK . ✄ Owing to Assumption ( i ) and to [ b ] S, + < + ∞ , ( x, y ) f ′ θ ( | x − y | )Ψ θ,S ( x, y )is always bounded from above on c ∆ R d × R d so that the integrals with respect to m ∈ P ⋆ are well-defined. Also note that since f ′ θ is positive, Assumption ( ii ) holds in particular ifthere exists θ and S such that the ( S, θ )-confluence function Ψ θ,S is negative on c ∆ R d × R d . ✄ If we also assume in ( a ), that (cid:16) t Z t P s ( x, dy ) ds (cid:17) t ≥ is tight then, so is (cid:16) t Z t Q s ( x, x ′ , dy, dy ′ ) ds (cid:17) t ≥ .Since, by construction, the weak limiting distributions of this sequence as t → + ∞ areinvariant distributions, it follows that 1 t Z t Q s ( x, x ′ , dy, dy ′ ) ds weakly converges to ν ∆ as t → + ∞ . This motivates the “weak confluence” terminology. ✄ It is natural to wonder if the assumptions for pathwise asymptotic confluence (claim( b )) are more stringent than Assumptions ( i ) and ( ii ). The fact that Ψ θ,S < R d × R d \ ∆ R d × R d implies Assumption ( ii ) has already been mentioned. One can also checksthat R e R v θ ( w ) w dw dv < + ∞ implies Assumption ( i ): one first derives from the Cauchycriterion that R e R v θ ( w ) w dw dv < + ∞ implies that R u u e R v θ ( w ) w dw dv → u → + . Usingthat v e R v θ ( w ) w dw is non-increasing on (0 , ue R u θ ( w ) w dw → + as u → R u θ ( w ) − w dw → −∞ and thus, Assumption ( i ). ✄ If b and σ are Lipschitz continuous, Kunita’s Stochastic flow theorem (see [Kun90],Section 4.5) ensures in particular that, if x = x , the solutions X x t and X x t a.s. neverget stuck. Taking advantage of this remark slightly shortens the proof below. ✄ Tightness criterions of (cid:16) t Z t P s ( x, dy ) ds (cid:17) t ≥ for every x ∈ R d usually rely on the mean-reversion property of the solutions of ( SDE ) usually established under various assumptionsinvolving the existence of a so-called
Lyapunov function V going to infinity at infinityand such that A V is upper-bounded and lim sup | x |→ + ∞ A V ( x ) < A denotes theinfinitesimal generator of X x (so-called Has’minskii’s criterion ). Keep in mind that A V ( x ) = ( b |∇ V )( x ) + 12 Tr (cid:16) σσ ∗ ( x ) D V ( x ) (cid:17) where Tr( A ) stands for the trace of the matrix A .On the other hand, a classical criterion for pathwise asymptotic confluence ( a.s. atexponential rate, see e.g. [BB92], [Lem05] and often referred to as asymptotic flatness ) is ∀ x, y ∈ R d , ( b ( x ) − b ( y ) | x − y ) + 12 k σ ( x ) − σ ( y ) k < − c | x − y | , c > , (3.13)14nd, as a straightforward consequence, uniqueness of the invariant distribution ν of ( SDE )(and of (
DSDS ) as well). Moreover, putting y = 0 in the above inequality straightfor-wardly yields real coefficients α > β ≥ A V ≤ β − αV with V ( x ) = | x | .Hence Has’minskii criterion is fulfilled, so it is also an existence criterion for the invari-ant distribution. In fact, both weak and pathwise assumptions in Theorem 3.2 are muchweaker than (3.13) but some of the properties which hold under (3.13) are still preserved.For instance, since the left-hand side of (3.13) corresponds to the ( S, S, c ∆ R d × R d , uniqueness of the invariant distribution ν of ( SDE ) (and of ν ∆ for ( DSDS ))holds and, combined with the tightness of the occupation measure of the semi-group, itbecomes a criterion for a.s. pathwise asymptotic confluence.
Proof of Theorem 3.2.
Step 1:
Exactly like in the beginning of the proof of Theo-rem 2.1( a ), one checks that the set I DSDS of invariant distributions of ( Q t ) t ≥ is anonempty, convex and weakly compact subset of P ( R d × R d ). As a a consequence ofthe Krein-Millman theorem, I DSDS has extremal distributions (and is their closed convexhull).On the other hand, it follows from strong uniqueness theorem for SDE’s that thesemi-group ( Q t ) t ≥ leaves stable the diagonal ∆ R d × R d = { ( x, x ) , x ∈ R d } .Let x , x ∈ R d , x = x . We define the stopping time τ x ,x := inf (cid:8) t ≥ | X x t = X x t (cid:9) . Still by a strong uniqueness argument it is clear that { τ x ,x > t } = { X x t = X x t } so that Q t (( x , x ) , c ∆ R d × R d ) = c ∆ R d × R d ( x , x ) P ( τ x ,x > t )and Q t (( x , x ) , c ∆ R d × R d ) = 0.Let µ ∈ I DSDS be an extremal invariant measure. We have, for every t ≥ µ ( c ∆ R d × R d ) = Z c ∆ R d × R d µ ( dx , dx ) P ( τ x ,x > t ) . Letting t go to + ∞ yields µ ( c ∆ R d × R d ) = Z c ∆ R d × R d µ ( dx , dx ) P ( τ x ,x = + ∞ )so that, on c ∆ R d × R d , µ ( dx , dx )- a.s. , P ( τ x ,x = + ∞ ) = 1 or equivalently the process( X x , X x ) lives in c ∆ R d × R d . Consequently, if µ ( c ∆ R d × R d ) ∈ (0 , µ c ∆ R d × R d and µ ∆ R d × R d are invariant distributions for ( SDSD ) as well. If so, µ = µ ( c ∆ R d × R d ) µ c ∆ R d × R d + µ (∆ R d × R d ) µ ∆ R d × R d cannot be extremal. Consequently µ (∆ R d × R d ) = 0 or 1. Step 2:
Let µ be an extremal distribution in I DSDS and assume that µ ( c ∆ R d × R d ) = 1so that µ ∈ P ⋆ . We will prove that this yields a contradiction under Assumptions ( i ) and( ii ).Note that f ′ θ and g θ defined in (3.10) are positive on (0 , + ∞ ), that Assumption ( i ) readslim sup u → + g θ ( u ) < + ∞ and that g ′ θ ( u ) = f ′ θ ( u )(1 − θ ( u )). Moreover, if Assumption ( ii )is fulfilled, so is the case for any continuous function e θ satisfying e θ ≥ θ . As a consequence,15e may modify θ on [1 , + ∞ ) so that θ still satisfies ( ii ) and θ ≥ ε, + ∞ ). Thenthe function g θ is non-increasing on [2 , + ∞ ). Consequently, without loss of generality, wemay assume in the sequel of the proof thatsup u> g θ ( u ) < + ∞ . (3.14)We now define a (Lyapunov) function ϕ : c ∆ R d × R d → R by ϕ ( y , y ) := f θ ( | y − y | S ) . We know from Step 1 that µ ( dx , dx )- a.s. , ( X x t = X x t for every t ≥ a.s. . Then, f θ being a C -function, we derive from Itˆo’s formula applied to ϕ ( X x t , X x t ) that µ ( dx , dx )- a.s. , ϕ ( X x t , X x t ) = ϕ ( x , x ) + Z t A (2) ϕ ( X x s , X x s ) ds + Z t f ′ θ ( | X x s − X x s | S ) (cid:0) ( σ ∗ ( X x s ) − σ ∗ ( X x s )) S ( X x s − X x s ) | dW s (cid:1)| {z } =: M t local martingale where, for every ( x , x ) ∈ ( R d ) , A (2) ϕ ( x , x ) = 2 (cid:16) ( b ( x ) − b ( x ) | x − x ) S + 12 k σ ( x ) − σ ( x ) k S (cid:17) f ′ θ ( | x − x | S )+ 2 f ′′ θ ( | x − x | S ) (cid:12)(cid:12) ( σ ∗ ( x ) − σ ∗ ( x )) S ( x − x ) (cid:12)(cid:12) . (3.15)Using that f θ is increasing and satisfies the ODE ≡ θ ( ξ ) f ′ θ ( ξ ) + ξf ′′ θ ( ξ ) = 0, ξ ∈ (0 , + ∞ ),we deduce that A (2) ϕ ( x , x ) = 2 f ′ θ ( | x − x | S )Ψ θ,S ( x , x )so that Z R d × R d A (2) ϕ ( x , x ) µ ( dx , dx ) < ii ).On the one hand, since µ is extremal and since A (2) ϕ is bounded from above (seeRemark 3.3), we can apply Birkhoff’s theorem and obtain: µ ( dx , dx )- a.s., t Z t A (2) ϕ ( X x s , X x s ) ds t → + ∞ −−−−→ Z c ∆ R d × R d A (2) ϕ dµ ∈ [ −∞ , a.s. (3.16)On the other hand, using that g θ is bounded and σ is Lipschitz continuous, it follows that( M t ) t ≥ is an L -martingale such that h M i t = Z t g θ ( | X x s − X x s | S ) (cid:12)(cid:12)(cid:12) ( σ ∗ ( X x s ) − σ ∗ ( X x s )) | X x s − X x s | S S ( X x s − X x s ) | X x s − X x s | S (cid:12)(cid:12)(cid:12) ds ≤ Ct P µ - a.s. (3.17)where C is a deterministic positive constant so that M t t → P µ - a.s. .As a consequence, µ ( dx , dx )- a.s. ,lim t → + ∞ ϕ ( X x t , X x t ) t = Z c ∆ R d × R d A (2) ϕdµ < a.s.. a.s. , f θ ( | X x t − X x t | S ) = ϕ ( X x t , X x t ) t → + ∞ −−−−→ −∞ a.s. If lim u → + f θ ( u ) > −∞ , this yields a contradiction since f θ is increasing on R ∗ + . Otherwise | X x t − X x t | S t → + ∞ −−−−→
0. But applying again Birkhoff’s theorem, we obtain µ ( dx , dx )- a.s. , Z | y − y | S µ ( dy , dy ) = lim t → + ∞ t Z t | X x t − X x t | S ds = 0 a.s., which contradicts the assumption µ ( c ∆ R d × R d ) = 1. Consequently, for any extremal invari-ant distribution µ , we have µ (∆ R d × R d ) = 1.We can now prove Claim ( a ): by Krein-Millman’s Theorem I SDS is the weak closure ofthe convex hull of its extremal distributions. Consequently, the diagonal ∆ R d × R d being aclosed subset of R d × R d , all invariant distributions of the duplicated system are supportedby this diagonal. For any such invariant distribution µ , both its marginals are invariantdistributions for ( SDE ). If (
SDE ) had two distinct invariant distributions ν and ν ′ , weknow from the introduction that I DSDS would contain at least a distribution µ for whichthe two marginals distributions are µ ( . × R d ) = ν and µ ( R d × . ) = ν ′ respectively. Asa consequence, such a distribution µ could not by supported by the diagonal ∆ R d × R d .Finally, I SDE is reduced to a singleton { ν } and I DSDS = { ν ∆ } . Step 3 (Claim ( b ) : Proof of (3.12) ): Under the additional assumption on θ of ( b ),we have lim u → + f θ ( u ) > −∞ and thus, inf u> f θ ( u ) > −∞ since f θ is increasing. Let x , x ∈ R d . Using again that A (2) ϕ < c ∆ R d × R d where A (2) ϕ is given by (3.15). It followsthat (cid:0) f θ ( | X x t − X x t | S ) (cid:1) t ≥ is a lower-bounded P -supermartingale. Thus, it a.s. convergestoward L x ,x ∞ ∈ L ( P ). Using again that f θ is increasing, it follows that | X x t − X x t | S a.s. converges toward a finite random variable ℓ x ,x ∞ = f − θ ( L x ,x ∞ ).Now, using that for every x ∈ R d , (cid:16) t Z t P s ( x, dy ) ds (cid:17) t ≥ is tight, we derive that (cid:16) t Z t Q s (( x , x ) , ( dy , dy )) ds (cid:17) t ≥ is tight as well. Then the uniqueness of ν ∆ as aninvariant distribution of Q implies that1 t Z t Q s (cid:0) ( x , x ) , ( dy , dy ) (cid:1) ds ( R d ) = ⇒ ν ∆ . Now for every bounded continuous function g : R d → R ,1 t Z t Q s ( g ( | y − y | S ))( x , x ) ds = 1 t Z t E g ( | X x s − X x s | S ) ds −→ E g ( ℓ x ,x ∞ )so that E g ( ℓ x ,x ∞ ) = Z g ( | y − y | S ) ν ∆ ( dy , dy ) = g (0) . ✷ In Assumption ( ii ) of the previous theorem, we see that the function ( x, y )
7→ | ( σ ∗ ( x ) − σ ∗ ( y )) S ( x − y ) | plays an important role. In the sequel, we will obtain specific results whenthis function is not degenerated away from the diagonal. Such type of assumption will becalled strong or regular directional S -ellipticity assumption .In the following proposition, we first show that when such an assumption is satis-fied, claim ( b ) of the previous theorem still holds without the tightness assumption on (cid:0) t R t P s ( x, dy ) ds (cid:1) t ≥ (although it is not really restrictive in our framework (see the fourthitem of Remark 3.3)). 17 ROPOSITION
If the function θ is (0 , -valued and σ satisfies the following strongdirectional S -ellipticity assumption away from the diagonal ∃ α > , ∀ x, y ∈ R d , (cid:12)(cid:12) ( σ ∗ ( x ) − σ ∗ ( y )) S ( x − y ) (cid:12)(cid:12) ≥ α | x − y | , (3.18) then the conclusion of Claim ( b ) in the above proposition remains true without the tightnessassumption on ( t R t P s ( x, dy ) ds ) t ≥ .Proof. First, we recall that under the assumptions of ( b ), we recall that ( f θ ( | X x t − X x t | S )) t ≥ is a lower-bounded P -supermartingale thus convergent to an integrable ran-dom variable and that this implies that ( | X x t − X x t | S ) t ≥ is a.s. convergent to a finiterandom variable ℓ x ,x ∞ (since f θ is increasing). On the other hand, since −A (2) ϕ is positiveand f θ is lower-bounded, we also have that f θ ( | X x t − X x t | S ) − Z t A (2) ϕ ( X x s , X x s ) ds = ϕ ( x , x ) + M t is a lower bounded P -(local) martingale starting at a deterministic starting value, henceconverging toward an integrable random variable. Owing to the computations of (3.17)(which hold for every starting points x , x ), ( M t ) t ≥ is in fact an L - convergent mar-tingale. Thus, h M i ∞ < + ∞ and taking advantage of the expression of this bracket (see(3.17)) and to Assumption (3.18), we derive that for every ε > Z + ∞ g θ (cid:0) | X x s − X x s | S (cid:1) {| X x s − X x s | S ≥ ε } ds < + ∞ a.s. The function g θ is positive on (0 , + ∞ ) and non-decreasing since g ′ θ ( u ) = f ′ θ ( u )(1 − θ ( u )) ≥
0. This implies that, for every ε > t → + ∞ g θ (cid:0) | X x t − X x t | S (cid:1) {| X x − X x | S ≥ ε } = 0 a.s. Combined with the convergence of the squared norm this yields ∀ ε > , g θ (cid:0) ℓ x ,x ∞ (cid:1) { ℓ x ,x ∞ ≥ ε } = 0 a.s. which finally implies ℓ x ,x ∞ = 0 a.s. In this section and the following, we derive several corollaries of Theorem 3.2 illustratedby different examples. P ROPOSITION
Let S ∈ S ++ ( d, R ) . Assume [ b ] S, + < + ∞ , σ is Lipschitz continuousand I SDS is non empty and weakly compact. Then, ( a ) If Assumption ( ii ) of Theorem 3.2 holds with some continuous functions θ : (0 , + ∞ ) → R + satisfying: there exists ε > such that θ ( u ) ≤ , u ∈ (0 , ε ] , then ( SDE ) (1.1) and itsduplicated system ( DSDS ) have ν and ν ∆ as unique invariant distributions respectively. ( b ) If for every x ∈ R d , ( t R t P s ( x, dy ) ds ) t ≥ is tight and if Ψ θ,S < on ∆ c R d × R d with acontinuous function θ : (0 , + ∞ ) → R + satisfying: there exists κ > and ε ∈ (0 , e − κ ) such that ∀ u ∈ (0 , ε ] , θ ( u ) ≤ (cid:16) κ log u (cid:17) , (3.19)18 hen the duplicated system of ( SDE ) is pathwise confluent in the sense of Theorem 3.2 ( b ) .This condition is in particular satisfied if there exists ε > and θ ∈ (0 , such that ∀ u ∈ (0 , ε ] , θ ( u ) ≤ θ . Proof.
Claim ( a ) is obvious. As for ( b ), one checks that R e R v θ ( w ) w dw dv < + ∞ as soon aslim inf u → + log( u ) (cid:0) θ ( u ) − (cid:1) > R EMARK . The simplest case where the preceding result holds is obtained when θ ≡ S, ∀ m ∈ P ⋆ , Z R d × R d ( b ( x ) − b ( y ) | x − y ) S + 12 k σ ( x ) − σ ( y ) k S m ( dx, dy ) < b ) holds as soon as the integrated function is negative on c ∆ R d × R d .At this stage, it is important for practical applications to note that the constant func-tion θ ≡ satisfies the assumption in ( a ) of the above Proposition . This leads us tointroduce an important quantity of interest for our purpose. D EFINITION
The non-infinitesimal S -Lyapunov (NILS) exponent is a function on R d × R d \ ∆ R d × R d defined for every x, y ∈ R d , x = y , byΛ S ( x, y ) = ( b ( x ) − b ( y ) | x − y ) S | x − y | S + 12 k σ ( x ) − σ ( y ) k S | x − y | S − (cid:12)(cid:12)(cid:12) ( σ ∗ ( x ) − σ ∗ ( y )) S ( x − y ) | x − y | S (cid:12)(cid:12)(cid:12) ! . C OROLLARY
Assume b and σ are like in Proposition 3.3 and I SDE is non emptyand weakly compact. ( a ) Negative Integrated NILS exponent : if ∀ m ∈ P ⋆ , Z R d Λ S ( x, y ) m ( dx, dy ) < , (3.21) then ( SDE ) and its duplicated system ( DSDS ) have ν and ν ∆ as unique invariant distri-butions respectively. ( b ) Negative NILS exponent bounded away from 0 : If furthermore (cid:0) t R t P s ( x, dy ) ds (cid:1) t ≥ is tight for every x ∈ R d or σ satisfies (3.18) and if there exists c > such that ∀ x, y ∈ R d , x = y, | x − y | S ≤ ε = ⇒ Λ S ( x, y ) ≤ − c (3.22) then the duplicated diffusion is pathwise confluent i.e. ∀ x , x ∈ R d , X x t − X x t −→ a.s. as t → + ∞ . Proof. ( a ) follows from claim ( a ) in the above proposition with θ ≡ uf ′ θ ( u ) ≡ , + ∞ ) so that A (2) ϕ ( x, y ) = 2Λ S ( x, y ) in the proof of Theorem 3.2. ( b ) follows fromclaim ( b ) in the same proposition. 19 EMARK . ✄ It is obvious that (3.21) is satisfied, i.e. the integrated NILS (INILS)exponent is negative for every m ∈ P ⋆ , as soon as the NILS exponent itself is negativeon c ∆ R d × R d . This pointwise negativity may appear as the only checkable condition forpractical applications, but so is not the case and we will see in the next subsections thatwe can devise criterions when Λ S is not negative everywhere. ✄ Let us assume that ν is unique, that ν ⊗ ν ∈ P ⋆ν,ν (for instance, so is the case if ν isatomless) and that b and σ are such that for all x = y , for al ν ⊗ ν ∈ P ⋆ν,ν (for instance,so is the case if ν is atomless) and that b and σ are such that for all x = y , for all t ≥ P ( X xt = X yt ) = 1 (see the fourth item of Remark 3.3 for comments on this topic). Then,for each t >
0, one easily checks that the probability measure µ (0) t defined in (1.4) belongsto P ⋆ν,ν . Furthermore, if (3.21) holds, ( µ (0) t ) t ≥ converges weakly to ν ∆ since this familyis weakly compact (see the first item of Remark 3.3) and ν ∆ is its only possible limitingdistribution. On the other hand, the function Λ S being continuous and upper-bounded on c ∆ R d × R d , can be extended on the diagonal into a lower semi-continuous (l.s.c.) functionon R d × R d (with values in R ∪ {−∞} ). Let Λ S denote its l.s.c. envelope. Temporarilyassume that b and σ are Lipschitz continuous so that Λ S is bounded. Then applyingFatou’s Lemma in distribution to Λ S , it follows from (3.21) that Z Λ S ( x, x ) ν ( dx ) = Z Λ S ( x, y ) ν ∆ ( dx, dy ) ≤ lim inf t → + ∞ Z Λ S ( x, y ) µ (0) t ( dx, dy ) ≤ . (3.23)The interesting point is that the left-hand side of (3.23) is an integral with respect to ν and can be seen as a necessary condition for the criterion (3.21). In fact the result stillholds if [ b ] S, + < + ∞ mutatis mutandis and there exists a continuous function ℓ ∈ L ( ν )such that ∀ x, y ∈ R d × R d , Λ − S ( x, y ) ≤ ℓ ( x ) + ℓ ( y ) (3.24)(where, for a function f , f ± = max( ± f, b and σ are continuously differentiable, one derives from a Laplace-Taylor expansion (integral remainder) that Λ S ( x, x ) reads for every x ∈ R d :Λ S ( x, x ) = 12 inf | u | S =1 (cid:18) u ∗ ( SJ b ( x ) + J ∗ b ( x ) S ) u + (cid:13)(cid:13)(cid:13) ( ∇ σ ( x ) | u ) (cid:13)(cid:13)(cid:13) S − (cid:12)(cid:12)(cid:12) ( ∇ σ ∗ ( x ) Su | u ) (cid:12)(cid:12)(cid:12) (cid:19) . (3.25)Thus, (3.23) can be read as a checkable necessary condition for the criterion (3.21). Wewill come back on this condition in Subsection 3.6. ✄ In ( b ), Condition (3.22) can be replaced by the sharper: for all x, y ∈ R d , Λ S ( x, y ) < κ > ε ∈ (0 , e − κ ) such that for all x, y ∈ R d such that | x − y | S ≤ ε ,Λ S ( x, y ) ≤ κ log( | x − y | S ) .When the coefficients are smooth enough, the negativity of Λ S can be ensured by thefollowing criterion: C OROLLARY (Smooth coefficients) . Assume the functions b and σ are continuously differentiable. Let J b ( x ) = h ∂b i ∂x j ( x ) i ≤ i,j ≤ d denote the Jacobian of b at x and let ∇ σ ( x ) = h ∂σ ij ∂x k ( x ) i i,j,k denote the gradient of σ at x . If both SJ b + J ∗ b S and ∇ σ are Lipschitzcontinuous and if ∇ σ is bounded then Λ S ( x, y ) ≤ − c on c ∆ R d × R d if sup x ∈ R d sup | u | S =1 (cid:18) u ∗ ( SJ b ( x ) + J ∗ b ( x ) S ) u + (cid:13)(cid:13)(cid:13) ( ∇ σ ( x ) | u ) (cid:13)(cid:13)(cid:13) S − (cid:12)(cid:12)(cid:12) ( ∇ σ ∗ ( x ) Su | u ) (cid:12)(cid:12)(cid:12) (cid:19) < here, for every v = ( v , . . . , v d ) ∈ R d , ( ∇ σ ( x ) | v ) = h ( ∇ σ ij ( x ) | v ) i ≤ i,j ≤ d and ∇ σ ( x ) v = " d X k =1 ∂σ ij ∂x k ( x ) v k ≤ i,j ≤ d . When S = I d , this may also be written sup x ∈ R d sup | u | =1 u ∗ (cid:16) ( J b + J ∗ b )( x ) + X i,j ( ∇ σ ij ( x )) ⊗ − h ( ∇ σ ij ( x ) | u ) ih ( ∇ σ ∗ ij ( x ) | u ) i(cid:17) u < . The proof is again an easy consequence of the Laplace-Taylor formula. Computationaldetails are left to the reader.
By local we mean that the confluence condition will be effective only in the neighbourhoodof the diagonal ∆ R d × R d . The price to pay is a regular directional ellipticity assumption on σ ( x ) − σ ( y ) in the direction S ( x − y ) away from the diagonal. P ROPOSITION
Assume [ b ] S, + < + ∞ , σ is Lipschitz continuous and ( SDE ) admitsat least one invariant distribution ν . If there exists ε > such that ( i ) Directional S -ellipticity: η := inf n(cid:12)(cid:12) ( σ ∗ ( x ) − σ ∗ ( y )) S ( x − y ) (cid:12)(cid:12) , | x − y | S ≥ ε o > , ( ii ) Locally negative INILS exponent: ∀ m ∈ P ⋆ , Z | x − y | S ≤ ε Λ S ( x, y ) m ( dx, dy ) < , then ( SDE ) (1.1) and its duplicated system still have ν and ν ∆ as unique invariant dis-tributions respectively.Proof. Owing to ( i ), we have for every u ∈ ( ε , + ∞ ):sup | x − y | S = u ( b ( x ) − b ( y ) | x − y ) S + k σ ( x ) − σ ( y ) k S (cid:12)(cid:12)(cid:12) ( σ ∗ ( x ) − σ ∗ ( y )) S x − y | x − y | S (cid:12)(cid:12)(cid:12) . ≤ (cid:0) [ b ] + + 12 [ σ ] (cid:1) u η . For every ε ′ > ε , let θ ε ′ : (0 , + ∞ ) → R + denote the continuous function defined by θ ε ′ ( u ) = (cid:0) [ b ] S, + + [ σ ] (cid:1) u η if u ∈ ( ε ′ , ∞ ) , u ∈ (0 , ε ] , θ ε ′ ( ε ′ ) t − ε ε ′ − ε if u ∈ ( ε , ε ′ ) . Since θ ε ′ ( u ) = 1 in the neighbourhood of 0, Assumption ( i ) of Theorem 3.2 is satisfied.For Assumption ( ii ), one first deduces from the construction and to the first assumptionthat, ∀ ε ′ > ε , ∀ x, y such that | x − y | S ≥ ε ′ , Ψ θ ε ′ ,S ( x, y ) ≤ . Using that θ ε ′ ( u ) = 1 on (0 , ε ], it follows that for all m ∈ P ⋆ , Z R d × R d f ′ θ ( | x − y | S )Ψ θ ε ′ ,S ( x, y ) m ( dx, dy ) ≤ Z | x − y | S ≤ ε Λ S ( x, y ) m ( dx, dy ) + I ( ε ′ )21here I ( ε ′ ) = Z ε < | x − y | S <ε ′ f ′ θ ( | x − y | S )Ψ θ ε ′ ,S ( x, y ) m ( dx, dy ) . Since the integrated function is bounded from above on { ( x, y ) , ε < | x − y | < ε ′ } , wededuce that I ( ε ′ ) ≤ C m ( ε < | x − y | < ε ′ ) ε ′ → ε −−−−→ . By the second assumption of this proposition, it follows that there exists ε ′ > ε such that Z R d × R d f ′ θ ( | x − y | S )Ψ θ ε ′ ,S ( x, y ) m ( dx, dy ) < ii ) of Theorem 3.2 holds. This completes the proof. ✷ As mentioned in Remark 3.4, Theorem 3.2 can be applied under the ( S, ∀ x, y ∈ R d , x = y, ( b ( x ) − b ( y ) | x − y ) S + 12 k σ ( x ) − σ ( y ) k S < . One asset of this more stringent assumption is that it can be localized in two ways:first in the neighbourhood of the diagonal like in the above local criterions, but also oncompacts sets of R d × R d . This naturally leads to a criterion based on the differentials of b and σ when they exist. P ROPOSITION (Criterion on compact sets) . ( a ) Let S ∈ S ++ ( d, R ) such that for every R > , there exists δ R > such that ∀ x, y ∈ B | . | S (0; R ) , < | x − y | S ≤ δ R = ⇒ ( b ( x ) − b ( y ) | x − y ) S + 12 k σ ( x ) − σ ( y ) k S < . (3.26) Then the diffusion is asymptotically ( S, -confluent. ( b ) If b and σ are continuously differentiable, then (3.26) holds as soon as ( AC ) diff ≡ ∀ x ∈ R d , SJ b ( x ) + J ∗ b ( x ) S + √ S X i,j ( ∇ σ ij ( x )) ⊗ √ S < in S ( d, R ) . Proof. ( a ) Let x, y ∈ R d such that x = y . Set R = max( | x | S , | y | S ) and x = x, x i = x + iN ( y − x ) , i = 1 , . . . , N − , x N = y where | y − x | S < N δ R . Then for every i ∈ { , . . . , N } , | x i | S ≤ R and | x i − x i − | S ≤ δ R .Then k σ ( x ) − σ ( y ) k S = (cid:13)(cid:13)(cid:13) N X i =1 σ ( x i ) − σ ( x i − ) (cid:13)(cid:13)(cid:13) S ≤ N N X i =1 k σ ( x i ) − σ ( x i − ) k S < − N N X i =1 ( b ( x i ) − b ( x i − ) | x i − x i − ) S = − N X i =1 (cid:0) b ( x i ) − b ( x i − ) | y − x (cid:1) S < − b ( y ) − b ( x ) | y − x ) S . b ) First, we prove the result when S = I d . We note that, for every continuouslydifferentiable function g : R d → R , g ( y ) − g ( x ) = R (cid:0) ∇ g ( x + t ( y − x )) | y − x (cid:1) dt = R ( y − x ) ∗ ∇ g ( x + t ( y − x )) dt so that (cid:0) b ( y ) − b ( x ) | y − x (cid:1) = Z ( y − x ) ∗ J b ( x + t ( y − x ) (cid:1) ( y − x ) dt = Z ( y − x ) ∗ J ∗ b ( x + t ( y − x ) (cid:1) ( y − x ) dt and k σ ( y ) − σ ( x ) k = d X i,j =1 (cid:18)Z (cid:0) ∇ σ ij ( x + t ( y − x )) | y − x (cid:1) dt (cid:19) . By Schwarz’s Inequality and the fact that ( u | v ) = u ∗ v ⊗ u , we deduce (cid:0) b ( y ) − b ( x ) | y − x (cid:1) + 12 k σ ( y ) − σ ( x ) k ≤ Z (cid:16) ( y − x ) ∗ ( J b + J ∗ b )( x + t ( y − x ))( y − x )+ 12 X ij ( y − x ) ∗ (cid:0) ∇ σ ij ( x + t ( y − x )) (cid:1) ⊗ ( y − x ) (cid:17) dt. This completes the proof when S = I d . This extends to general matrix S ∈ S ++ ( d, R )using that k σ ( y ) − σ ( x ) k S = k ( √ Sσ )( y ) − ( √ Sσ )( x ) k and the fact that ( Au ) ⊗ = Au ⊗ A ∗ with A = √ S . ✷ Λ S ≤ S < R d × R d . In the two next sections, our objective is to statesome results when this condition is not fulfilled. We begin by a simple application ofCorollary 3.2 where the NILS exponent is only non-positive and negative outside of acompact set. P ROPOSITION
Assume [ b ] S, + < + ∞ , σ is Lipschitz continuous and ( SDE ) has aunique invariant distribution ν whose support is not compact. Then, uniqueness for ν ∆ holds true as soon as ∀ x, y ∈ R d , Λ S ( x, y ) ≤ ∃ R > s.t. max( | x | S , | y | S ) > R = ⇒ Λ S ( x, y ) < . (3.27) Proof.
Since the support of ν is not compact, we have for every m ∈ P ⋆ν,ν : m (cid:0) { max( | x | S , | y | S ) >R } (cid:1) ≥ ν ( {| x | S > R } ) >
0. It follows from the assumption that ∀ m ∈ P ⋆ν,ν , Z Λ S ( x, y ) m ( dx, dy ) ≤ Z Λ S ( x, y )1 { max( | x | S , | y | S ) >R } m ( dx, dy ) < R EMARK . ✄ In the particular case where σ is constant, Condition (3.27) becomes amonotony condition on b (decrease with respect to ( . | . ) S at infinity), namely: ∀ x, y ∈ R d , x = y ( b ( x ) − b ( y ) | x − y ) S ≤ , and ∃ R > | x | S , | y | S ) > R = ⇒ ( b ( x ) − b ( y ) | x − y ) S < . b is S -non-increasing on R d , S -decreasing outside B | . | S (0; R ) . For in-stance, if b = −∇ U , the above assumption holds if U is convex and (only) strictly convexoutside of a compact set. ✄ Note that when ∇ U is only increasing outside B | . | S (0; R ) but possibly with no specificmonotony on B | . | S (0; R ), it is still possible to find some diffusion coefficients σ such thatthe SDE dX t = −∇ U ( X t ) dt + σ ( X t ) dW t remains weakly or pathwise confluent. We refer tothe next subsection for models with such stochastically stabilizing diffusive components. ✄ Finally, note that the above condition (3.27) can be also localized around the diagonalunder the directional S -ellipticity assumption. To be more precise, when ν is unique andits support is not compact, Proposition 3.6 still holds if Assumption ( ii ) is “localized”into: ( ii ) loc ≡ for every x, y ∈ R d such that 0 < | x − y | ≤ ε , Λ S ( x, y ) ≤ . Λ S possibly positive on some areas of R d × R d In the continuity of the previous section, we try to explore some multidimensional settingswhere Λ S can be positive in some parts of the space. More precisely, we focus here ongradient systems with constant noise whose potential U is not convex in all the space (see[Tea08] for other confluence results on this type of model with the “random attractor”viewpoint). For such dynamical systems, we obtain a criterion below that we next applyto super-quadratic non-convex potentials. Then, we will come back to this problem insection 3.5.3 where we focus on the particular example U ( x ) = ( | x | − , case for whichwe are able to obtain a sharper result. P ROPOSITION (Gradient system) . Let U : R d → R + be a locally Lipschitz, differen-tiable function satisfying < lim inf | x |→ + ∞ U ( x ) | x | γ < lim sup | x |→ + ∞ U ( x ) | x | γ < + ∞ for a positive γ . Then, the Brownian diffusion dX xt = −∇ U ( X xt ) dt + σdW t , X x = x, where σ > and W is a standard Brownian motion on R d , satisfies a strong existence-uniqueness property with unique invariant distribution ν σ ( dx ) = C σ e − U ( x ) σ dx .Furthermore assume that its NILS exponent satisfies ∀ x, y ∈ R d , Λ Id ( x, y ) ≤ β − α (cid:0) | x | a + | y | a (cid:1) where β ∈ R , α, a > then there exists σ c > such that, for every σ > σ c , the related ( DSDS ) system( -point motion) is weakly confluent.Proof. The strong existence-uniqueness is classical background. The form of the invariantdistribution ν σ as well. Then by Fatou’s Lemma and the asymptotic upper-bound, thereexists A > σ → + ∞ Z R d | u | a e − U ( σ /γu ) σ du ≥ Z R d | u | a e − A | u | γ du > . On the other hand, note that Z R d | x | a e − U ( x ) σ dx = σ d + aγ Z R d | u | a e − U ( σ /γu ) σ du. U at infinity and the (reverse) Fatou’s Lemma,there exists a real number B > σ → + ∞ Z R d e − U ( σ /γu ) σ du ≤ Z R d e − B | u | γ du < + ∞ . As a consequence lim inf σ → + ∞ ν σ (cid:0) | x | a (cid:1) = + ∞ . For any distribution m ∈ P ( R d × R d ) withmarginal ν σ and assigning no weight to the diagonal, one has Z R d × R d \ ∆ R d × R d Λ Id ( x, y ) m ( dx, dy ) ≤ β − αν σ (cid:0) | x | a (cid:1) < σ is large enough to ensure that ν σ (cid:0) | x | a (cid:1) ≥ βα . R EMARK . We may assume without loss of generality that argmin R d U = { U = 0 } ⊂{∇ U = 0 } so that ν σ R d = ⇒ ν = Unif( { U = 0 } ) as σ →
0. Hence, from a practical pointof view, the fact that the critical σ c can be taken as 0 seems a reasonable conjecture if β − αν (cid:0) | x | a { U ( x )=0 } (cid:1) ≤
0. Thus, in Section 3.5.3, we prove that it holds true for thepotential fonction U ( x ) = ( | x | − . C OROLLARY
Assume that U : R d → R + is defined by U ( x ) = C | x | p + ε ( x ) where p > , C > and ε is a C -function such that ∇ ε is Lipschitz continuous. Then, thereexists σ c > such that, for every σ > σ c , the ( DSDS ) related to the gradient system dX t = −∇ U ( X t ) dt + σdW t is weakly confluent.Proof. Using that for every x ∈ R d (even if x = 0 with an obvious extension by continuity), D ( | x | p ) = 2 p | x | p − (cid:18) p − x ⊗ | x | + I d (cid:19) ≥ p | x | p − I d in S + ( d, R ) , we deduce that for every x = y , (cid:0) ∇ ( | x | p ) − ∇ ( | y | p ) | x − y (cid:1) | x − y | ≥ p Z | y + t ( x − y ) | p − dt. If p ≥
2, we deduce from Jensen’s inequality that Z | y + t ( x − y ) | p − dt ≥ (cid:18)Z | y + t ( x − y ) | dt (cid:19) p − ≥ (cid:18)
16 ( | x | + | y | ) (cid:19) p − ≥ (cid:18) (cid:19) p − (cid:0) | x | p − + | y | p − (cid:1) where in the last inequality, we used again that p − ≥
1. It follows thatΛ Id ( x, y ) ≤ [ ε ] − α p ( | x | p − + | y | p − )where [ ε ] denotes the Lipschitz constant of ε and α p >
0. The previous result then appliesin this case.When p ∈ (1 , || u | ρ − | v | ρ | ≤ | u − v | ρ for0 < ρ < Z | y + t ( x − y ) | p − dt ≥ α p (cid:0) | x | p − + | y | p − (cid:1) with α p > .5 Examples Assume that σ : R d → M ( d, d, R ) is defined by σ ( x ) = x ⊗ λ + σ where σ ∈ M ( d, d, R )and λ : R d → R d is a bounded Lipschitz function (such that σ is Lipschitz too). If thereexists ρ ∈ (0 , ) such that lim sup | x |→ + ∞ ( b ( x ) | x ) − ρ | x | | λ ( x ) | (1 + | x | ) ρ + = −∞ (3.29)then the diffusion (1.1) has at least one invariant distribution ν . Thus, if[ b ] = sup x =0 ( b ( x ) − b (0) | x ) | x | = + ∞ , the above condition is satisfied as soon aslim inf | x |→ + ∞ | λ ( x ) | > b ] . The key is to introduce the Lyapunov function V ( x ) = ( a + | x | ) ρ + . Using that k ( σ − σ )( x ) k = | ( σ − σ ) ∗ ( x ) x | x | | = | λ ( x ) | | x | , we deduce that12 k σ ( x ) k − ( ρ + 12 ) (cid:12)(cid:12)(cid:12) σ ∗ ( x ) x | x | (cid:12)(cid:12)(cid:12) = − ρ | λ ( x ) | | x | + O (1)and it follows that lim sup | x |→ + ∞ A V ( x ) = −∞ if (3.29) is fulfilled (where A denotes theinfinitesimal generator of (1.1)).If the function λ is constant , the diffusion is asymptotically pathwise confluent (so that ν is unique for (1.1) and the duplicated system has ν ∆ as unique invariant distribution)as soon as there exists ε > | x − y | ≤ ε = ⇒ ( b ( x ) − b ( y ) | x − y ) − | λ | | x − y | < . (3.30)This is a consequence of Proposition 3.4 applied with S = I d (the directional ellipticityassumption ( i ) is clearly true since | ( σ ∗ ( x ) − σ ∗ ( y ))( x − y ) | = | λ | . | x − y | ). If b is smooththis condition is satisfied as soon as, for every x ∈ R d , ( J b + J ∗ b )( x ) < | λ | I d in S ( d, R )( J b ( x ) denotes the Jacobian matrix of b ). Let Ξ t = ( X t , Y t ) be the unique strong solution to the 2-dimensional SDE dX t = (cid:0) a − σ (cid:1) X t dt − ( σY t − θ X ) dW t dY t = (cid:0) b − σ (cid:1) Y t dt + ( σX t + θ Y ) dW t where W is scalar standard Brownian motion, a , b , σ are real numbers satisfying ab < , a + b < , σ > r aba + b . θ X , θ Y ∈ R . When θ X = θ Y = 0, this system is known as Baxendale’s system(see e.g. [KP92]). Its stochastic stability has been extensively investigated in connectionwith its Lyapunov exponent. Then set λ = λ ( σ ) = b − a + p ( b − a ) + σ σ ∈ (0 ,
1) and α = σ − ( a + b ) − p ( a − b ) + σ > . and | . | λ = | . | S with S = Diag(1 , λ ). Itˆo’s Lemma implies d | Ξ t | λ = (cid:16) − α | Ξ t | λ + θ X ( θ X − σY t ) + λθ Y ( θ Y + 2 σX t ) (cid:17) dt + Θ(Ξ t ) dW t where Θ( x, y ) = 2 (cid:0) ( λ − σxy + λθ Y x + θ X y (cid:1) . It is clear that there exists β ∈ R + suchthat | θ X ( θ X − σy ) + λθ Y ( θ Y + 2 σx ) | ≤ β ( | ( x, y ) | λ + 1) . Then using that | ξ | λ ≤ α + α | ξ | λ and setting β ′ = β + α , we derive that d | Ξ t | λ ≤ β ′ − α | Ξ t | λ dt + Θ(Ξ t ) dW t where θ ( ξ ) ≤ C | ξ | λ . Hence, the function V ( ξ ) = | ξ | λ is a Lyapunov function for the systemsince AV ≤ β ′ − α V . As a consequence there exists at least one invariant distribution ν for the system and any such distribution satisfies ν ( V ) ≤ β ′ α .At this stage we can compute the non-infinitesimal S -Lyapunov exponent of the du-plicated system. Tedious although elementary computations show that, for every ξ =( x, y ) , ξ ′ = ( x ′ , y ′ ) ∈ R ,Λ S ( ξ, ξ ′ ) = − α − ( λ − σ ( x − x ′ ) ( y − y ′ ) | ξ − ξ ′ | λ < . R EMARK . Adapting results from [Bax91] obtained for diffusions on compact man-ifolds, one easily derive another type of criterion for weak confluence. Namely, if thediffusion ( X xt ) t ≥ (1-point motion) has a unique invariant distribution ν with support R d and if X xt L −→ ν as t → + ∞ for every x ∈ R d and if no nonempty closed connected subset C of R d × R d \ ∆ R d × R d is left stable by the -point motion ( X x t , X x t ) t ≥ (( x , x ) ∈ C ), thenthe 2-point motion is weakly confluent with invariant distribution ν ∆ = ν ◦ ( x ( x, x )) − .However, although more intuitive this criterion seems not to be tractable compared to theabove criterions based on the N ILS exponent.
Let U : R d → R + be defined by U ( x ) = ( | x | − , x ∈ R d . Applying Corollary 3.4 with p = 2 and ε ( x ) = (1 − x ), one deduces that there exists σ c > σ > σ c , the 2-point motion related to dX xt = −∇ U ( X xt ) dt + σdW t is weakly confluent. Infact, for this function, we obtain the weak confluence for every σ > P ROPOSITION
Let U : R d → R + be defined by U ( x ) = ( | x | − . Then, for every σ > , the (DSDS) related to the Brownian diffusion dX xt = −∇ U ( X xt ) dt + σdW t is weaklyconfluent. roof. Elementary computations show that, for every x, y ∈ R d ,Λ Id ( x, y ) = 1 − (cid:16)(cid:0) | x | + | y | (cid:1) + ( x + y | x − y ) | x − y | (cid:17) ≤ − (cid:0) | x | + | y | (cid:1) so that for every m ∈ P ⋆ν,ν , Z Λ Id ( x, y ) m ( dx, dy ) < − (cid:18)Z | x | ν σ ( dx ) + Z | y | ν σ ( dy ) (cid:19) = 1 Z σ Z (1 − | x | ) e − U ( x ) σ dx (3.31)with Z σ = Z R d e − U ( x ) σ dx . By Corollary 3.2( a ), it is now enough to prove that Z R d (1 − | x | ) e − U ( x ) σ ( dx ) < . Thanks to a change of variable, Z R d (1 − | x | ) e − U ( x ) σ dx = Vol( S d − ) Z + ∞ (1 − r ) r d − e − ( r − σ dr where Vol( S d − ) denotes the hyper-volume of the d − d = 2, it follows that Z + ∞ (1 − r ) re − ( r − σ dr = σ (cid:20) e − ( r − σ (cid:21) + ∞ = σ e − σ < . When d >
2, note that (1 − r ) r d − ≤ (1 − r ) r for every r ∈ [0 , + ∞ ) so that Z + ∞ (1 − r ) r d − e − ( r − σ dr = Z + ∞ (1 − r ) re − ( r − σ dr < . This completes the proof.
As a conclusion of this first part of the paper, let us note that when ν is unique, the questionof the negativity of the Integrated NILS exponent on the set of probabilities m ∈ P ⋆ν,ν isconnected with an optimal transport problem (see e.g. [Vil09] for a background on thistopic).Let us be more precise. Assume that Λ S satisfies (3.24) and let ¯Λ S : R d × R d → R denote its upper semi-continuous (u.s.c.) envelope. If [ b ] S, + < + ∞ and σ is Lipschitzcontinuous, ¯Λ S is [ −∞ , C b,σ ]-valued where C b,σ is a real constant (note that when b and σ are continuously differentiable, the extension on the diagonal has an explicit form ob-tained by replacing the infimum by a supremum in (3.25)). If we slightly strengthen ourcriterion (3.21) – negativity of the the INILS exponent on P ⋆ν,ν – by also asking that R R d ¯Λ S ( x, x ) ν ( dx ) < ) and if we denote by P ν,ν ( R d × R d ) the (convex) set of distri-butions on R d × R d with marginals ν on R d , one checks that the more stringent resultingcriterion reads ∀ m ∈ P ν,ν ( R d × R d ) , Z R d × R d ¯Λ S ( x, y ) m ( dx, dy ) < . to be compared to the necessary condition (3.23). P ν,ν ( R d × R d ) and to the (weak) u.s.c. of the mapping m R R d × R d ¯Λ S ( x, y ) m ( dx, dy ), the above criterion is equivalent tomax (cid:26)Z R d × R d ¯Λ S ( x, y ) m ( dx, dy ) , m ∈ P ν,ν ( R d × R d ) (cid:27) < . Thanks to the Kantorovich duality Theorem and the symmetry of ¯Λ S , this criterion is inturn equivalent toinf (cid:26)Z R d ϕ dν, ϕ ∈ L ( ν ) , ϕ ( x )+ ϕ ( y ) ≥ ¯Λ S ( x, y ) , ( x, y ) ∈ R d × R d (cid:27) < . Note that this last formulation of the problem is well-posed since it only involves themarginal invariant distribution ν . For instance, it could be the starting point to devisingnumerical methods for testing the weak confluence of the diffusion.Note that the argument derived from (3.31) can be viewed as a duality-type argumentapplied with ϕ ( x ) = (1 − | x | ) and, more generally, so is the case for the criterion (3.28)in Proposition 3.7. As an application, we investigate in this section the Richardson-Romberg ( RR ) extrapo-lation for the approximation of invariant measures. Roughly speaking, the aim of a RR method is generally to improve the order of convergence of an algorithm based on andiscretization scheme by cancelling the first order error term induced by the time dis-cretization of the underlying process. However, to be efficient, such a method must beimplemented with a control of its variance. We will see that in this context, this control isstrongly linked to the uniqueness of the invariant distribution of the duplicated diffusion. Following [LP02] and a series of papers cited in the introduction, we consider here asequence of empirical measures ( ν ηn ( ω, dx )) n ≥ built as follows: let ( γ n ) n ≥ denote a non-increasing sequence of positive step parameters satisfying γ n n → + ∞ −−−−−→ n = n X k =1 γ k n → + ∞ −−−−−→ + ∞ . We denote by ( ¯ X n ) n ≥ the Euler scheme with step sequence ( γ n ) n ≥ defined by ¯ X = x ∈ R d and ¯ X n +1 = ¯ X n + γ n +1 b ( ¯ X n ) + √ γ n +1 σ ( ¯ X n ) U n +1 where ( U n ) n ≥ is a sequence of i.i.d. centered R q -valued random vectors such that Σ U = I q defined on a probability space (Ω , A , P ). The sequence of weighted empirical measures ( ν ηn ( ω, dx )) n ≥ is then defined for every n ≥
1, by ν ηn ( ω, f ) = 1 H n n X k =1 η k δ ¯ X k − ( ω ) δ a denotes the Dirac mass at a ∈ R d and ( η k ) k ≥ is a sequence of positive weightssuch that H n = P nk =1 η k n → + ∞ −−−−−→ + ∞ . When η k = γ k which corresponds to the genuinecase, we will only write ν n ( ω, dx ) instead of ν γn ( ω, dx ). For this sequence, we recall inProposition 4.9 below in a synthesized form the main convergence results (including rates)of the sequence ( ν ηn ( ω, dx )) to the invariant distribution ν of ( X t ). In this way, we introducetwo assumptions:( S a ) : ( a >
0) There exists a positive C -function V : R d → R withlim | x |→ + ∞ V ( x ) = + ∞ , |∇ V | ≤ CV, and sup x ∈ R d k D V ( x ) k < + ∞ such that there exist some positive constants C b , β and α such that: (i) | b | ≤ C b V a , Tr( σσ ∗ )( x ) = o ( V a ( x )) as | x | → + ∞ (ii) ( ∇ V | b ) ≤ β − αV a . This Lyapunov-type assumption is sufficient to ensure the long-time stability of the Eulersheme (in a sense made precise below) as soon as a ∈ (0 , a = 0 (see [Pan06]). The second assumption below is fundamental to establish the rateof convergence of ( ν ηn ( ω, f )) to ν ( f ) for a fixed smooth enough function f : R d → R : weassume that f has a smooth solution to the Poisson equation (see [PV01] for results onthis topic).( C ( f , k )): There exists a C k -function g : R d → R solution to f − ν ( f ) = A g such that f , g and its partial derivatives up to k are dominated by V r ( r ≥ | f | ≤ CV r and for every α = ( α , . . . , α d ) ∈ N d with | α | := α + · · · + α d ∈ { , . . . , k } , | ∂ | α | x α i ,...,x αdid g | ≤ CV r .Before recalling the results on ( ν n ( ω, dx )), let us introduce further notations. We set ∀ r ∈ N , Γ ( r ) n = n X k =1 γ rk and for a smooth enough function h : R d → R and an integer r ≥
2, we write: D ( r ) h ( x ) y ⊗ · · · ⊗ y r = X ( i ,...,i r ) ∈{ ,...,d } r ∂ rx i ,...,x ir h ( x ) y i . . . y i r r . P ROPOSITION
Assume ( S a ) holds for an a ∈ (0 , and U ∈ ∩ p> L p ( P ) . Assumethat ( η k /γ k ) is a non-increasing sequence. Then, ( i ) For every non-increasing sequence ( θ n ) n ≥ such that P n ≥ θ n γ n < + ∞ and for every r > , P n ≥ θ n γ n E [ V r ( ¯ X n )] < + ∞ . ( ii ) For every r > , sup n ≥ ν ηn ( ω, V r ) < + ∞ a.s. In particular, ( ν ηn ( ω, dx )) n ≥ is a.s. tight. ( iii ) Every weak limit of ( ν ηn ( ω, dx )) n ≥ is an invariant distribution for ( X t ) t ≥ . Further-more, if ( SDE ) has a unique invariant distribution, say ν , then ν ηn ( ω, f ) n → + ∞ −−−−−→ ν ( f ) a.s. for every ν - a.s continuous function f such that | f | ≤ CV r for an r > . ( iv ) (Rate of convergence when η k = γ k ): Assume that ν is unique and that E [ U ⊗ ] = 0 .Let k ≥ such that f : R d → R satisfies ( C ( f , k )) . Then, • If k = 4 and Γ (2) n √ Γ n n → + ∞ −−−−−→ , p Γ n ( ν n ( ω, f ) − ν ( f )) ( R ) = ⇒ N (cid:16) Z R d | σ ∗ ∇ g | dν (cid:17) as n → + ∞ . If k = 5 and Γ (2) n √ Γ n n → + ∞ −−−−−→ e β ∈ (0 , + ∞ ] , ✄ √ Γ n (cid:16) ν n ( ω, f ) − ν ( f ) (cid:17) ( R ) = ⇒ N (cid:16) e β m (1) g ; Z R d | σ ∗ ∇ g | dν (cid:17) as n → + ∞ if e β ∈ (0 , + ∞ ) , ✄ Γ n Γ (2) n ( ν n ( ω, f ) − ν ( f )) a.s. −→ m (1) g as n → + ∞ if e β = + ∞ where m (1) g = Z R d ϕ dν with ϕ ( x ) = 12 D g ( x ) b ( x ) ⊗ + 12 E [ D g ( x ) b ( x )( σ ( x ) U ) ⊗ ] + 124 E [ D g ( x )( σ ( x ) U ) ⊗ ] . (4.32)The first three claims part ( i ), ( ii ) and ( iii ) of the theorem follow from [LP03] whereasthe ( iv ) is derived from [LP02] (see Theorem 10) and [Lem05] (see Theorem V.3), in whichthe rate of convergence is established for a wide family of weights ( η k ).Applying ( iv ) to polynomial steps of the following form: γ n = Cn − µ , µ ∈ (0 , n − / and is attained for µ = 1 /
3. Then e β = √ C and p Γ n ∼ p C/ n . so that n (cid:16) ν n ( ω, f ) − ν ( f ) (cid:17) ( R ) = ⇒ N (cid:16) C m (1) g ; 23 C Z R d | σ ∗ ∇ g | dν (cid:17) . This corresponds to the case where the rate of convergence of the underlying diffu-sion toward its steady regime ( √ Γ n corresponding to √ t in the continuous time setting,see [Bha82] for the CLT for the diffusion itself) and the discretization error are of the sameorder. From a practical point of view it seems clear that a balance should be made be-tween the asymptotic bias and the asymptotic variance to specify the constant C . Underslightly more stringent assumptions we prove that the L –norm of the error ν n ( ω, f ) − ν ( f )satisfies k ν n ( ω, f ) − ν ( f ) k L ∼ n − s C ( m (1) g ) + 23 C Z R d | σ ∗ ∇ g | dν. An optimisation with respect to C gives the optimal choice C = R R d | σ ∗ ∇ g f | dν ( m (1) gf ) ! .When µ ∈ (0 , / m (1) g in order to extend the range of applicationof the rate √ Γ n (which corresponds to the standard weak rate √ t in Bhattacharia’s CLT )to “slower steps”.
As mentioned before, the starting idea is to introduce a second Euler scheme with stepsequence ( e γ n ) n ≥ defined by ∀ n ≥ , e γ n − = e γ n = γ n .
31s concerns the white noise of both schemes, our aim is to make them consistent inabsolute time and correlated (with correlation matrix ρ satisfying I q − ρ ∗ ρ ∈ S + ( d, R )).To achieve that we proceed as follows.Let ( Z n ) n ≥ be a sequence of i.i.d. R q -valued random vectors lying in ∩ p> L p ( P ) andsatisfying E Z = 0 , Σ Z = I q , E [ Z ⊗ ] = E [ Z ⊗ ] = 0 . Then we devise from this sequence the white noise sequence ( U n ) n ≥ of the “original”Euler scheme with step ( γ n ) n ≥ by setting ∀ n ≥ , U n = 1 √ Z n − + Z n ) . (4.33)The white noise sequence for the second Euler scheme (with step ( e γ n ) n ≥ ), denoted Z ( ρ ) is defined as follows: Z ( ρ ) n = ρ ∗ Z n + T ( ρ ) V n , n ≥ , (4.34)where ( V n ) n ≥ is also a sequence of i.i.d. centered random variables in R q with momentsof any order satisfying Σ V = I q and E [ V ⊗ ] = E [ V ⊗ ] = 0, independent of ( Z n ) n ≥ and T q ( ρ ) is a solution to the equation T q ( ρ ) T q ( ρ ) ∗ = I q − ρ ∗ ρ ∈ S + ( d, R ) . ( T q ( ρ ) can be chosen either as the commuting symmetric square root of I q − ρ ∗ ρ or itsCholeski transform). Note that ( Z ( ρ ) n ) n ≥ is built in so that it satisfiesΣ Z ( ρ ) n = I q and Cov( Z n , Z ( ρ ) n ) = ρ. Then the Euler scheme with step e γ n and consistent ρ -correlated white noise ( Z ( ρ ) n ) n ≥ ,denoted ( ¯ Y ( ρ ) n ) n ≥ from now on, is defined by:¯ Y ( ρ ) n +1 = ¯ Y ( ρ ) n + e γ n b ( ¯ Y ( ρ ) n ) + pe γ n σ ( ¯ Y ( ρ ) n ) Z ( ρ ) n +1 , n ≥ , ¯ Y = y. Also note that ( ¯ X n , ¯ Y ( ρ )2 n ) is an Euler scheme at time Γ n of the duplicated diffusion( X t , X ( ρ ) t ) t ≥ .For numerical purpose, one usually specifies the independent i.i.d. sequences ( Z n ) n ≥ and ( V n ) n ≥ as normally distributed so that they can be considered as the normalizedincrements of two independent Brownian motions W and f W i.e.Z n = W e Γ n − W e Γ n − pe γ n and V n = f W e Γ n − f W e Γ n − pe γ n , n ≥ . Note that in this case, ( U n ) is also a sequence of N (0 , I q )-random variables. This impliesin particular that E [ U ⊗ ] = E [ Z ⊗ ] and E [ U ⊗ ] = E [ Z ⊗ ] . (4.35)Since these properties simplify the result, we will assume them in the sequel of this section(see Remark 4.9 for extensions). 32e denote ( ν η, ( ρ ) n ( ω, dx )) n ≥ the sequence of empirical measures related to ( ¯ Y ( ρ ) n ( ω )) n ≥ (in which the weights are adapted accordingly: η / , η / , η / , η / , η / , . . . ). The em-pirical measure (¯ ν η, ( ρ ) n ( ω, dx )) n ≥ associated to the Richardson-Romberg extrapolation isdefined by ν η, ( ρ ) n ( ω, f ) = 1 H n n X k =1 η k (cid:16) f ( ¯ Y ( ρ )2( k − ( ω )) + f ( ¯ Y ( ρ )2 k − ( ω )) (cid:17) ¯ ν η, ( ρ ) n ( ω, f ) = (2 ν η, ( ρ ) n − ν ηn ( ω, f ))= 1 H n n X k =1 η k (cid:16) f ( ¯ Y ( ρ )2( k − ( ω )) + f ( ¯ Y ( ρ )2 k − ( ω )) − f ( ¯ X k ( ω )) (cid:17) . Under the assumptions of Proposition 4.9, it is clear that ¯ ν η, ( ρ ) n ( ω, dx ) n → + ∞ −−−−−→ ν ( dx ) a.s. .Thus, in the next section, we propose to evaluate the effects of the Richardson-Rombergextrapolation on the rate of convergence of the procedure and to explain why the unique-ness of the invariant distribution of the duplicated diffusion plays an important role inthis problem. Throughout this section we assume that η k = γ k and so we will write ν n , ν ( ρ ) n and ¯ ν ( ρ ) n in-stead of ν ηn , ν η, ( ρ ) n and ¯ ν η, ( ρ ) n respectively. We also set ( D g i,.,. ) di =1 = D ( ∇ . ) g in order thatthe notation Tr( σ ∗ D ( ∇ . ) gσ ) stands for the vector of R d defined by Tr( σ ∗ D ( ∇ . ) gσ ) =(Tr( σ ∗ D ( ∂ x i g ) σ )) di =1 . For a fixed matrix ρ , the main result about the RR extrapolationis Theorem 4.3 below. At this stage, we do not discuss the choice of the correlation ρ inthis result. This point is tackled in Proposition 4.10 in which we will see that the optimalchoice to reduce the asymptotic variance is atteined with ρ = I q as soon as ν ∆ is theunique invariant distribution of the associated duplicated diffusion. This emphasizes theimportance of the question of the uniqueness of the invariant distribution in this pathologiccase studied in the previous part of the paper. T HEOREM
Assume ( S a ) holds for an a ∈ (0 , . Assume that ( X t , X ( ρ ) t ) t ≥ admitsa unique invariant distribution µ ( ρ ) (with marginals ν ). Let f : R d → R be a functionsatisfying ( C ( f , )) and such that ϕ defined by (4.32) satisfies ( C ( ϕ , )) with a solutionto the Poisson equation denoted by g ϕ . Then, • If Γ (3) n √ Γ n n → + ∞ −−−−−→ , p Γ n (cid:16) ν ( ρ ) n ( ω, f ) − ν ( f ) (cid:17) n → + ∞ = ⇒ N (cid:0)
0; ˆ σ ρ (cid:1) where ˆ σ ρ = 5 Z R d | σ ∗ ∇ g | dν − Z R d × R d (cid:0) ( σ ∗ ∇ g )( x ) | ρ ( σ ∗ ∇ g )( y ) (cid:1) µ ( ρ ) ( dx, dy ) . (4.36) • If Γ (3) n √ Γ n n → + ∞ −−−−−→ e β ∈ (0 , + ∞ ] , then √ Γ n (cid:0) ν ( ρ ) n ( ω, f ) − ν ( f ) (cid:1) ( R ) = ⇒ N (cid:0) e β m (2) g ; ˆ σ ρ (cid:1) as n → + ∞ if e β ∈ (0 , + ∞ ),Γ n Γ (3) n ( ν ( ρ ) n ( ω, f ) − ν ( f )) P −→ m (2) g as n → + ∞ if e β = + ∞ ,33 here m (2) g = 12 (cid:18) m g ϕ + Z R d ϕ dν (cid:19) with ϕ ( x ) = X k =3 C k − k k ! E (cid:2) D k g ( x ) b ( x ) ⊗ (6 − k ) ( σ ( x ) U ) ⊗ k − (cid:3) . (4.37) R EMARK . ✄ We recall that the result is stated under the assumption that the in-crements are normally distributed or more precisely under Assumption (4.35). When thisadditional assumption fails (think for instance to Z ∼ (cid:0) ( δ − + δ ) (cid:1) ⊗ q ), the result isremains true except for the value of m (2) g which becomes more complicated since it alsodepends on E [ Z ⊗ ℓ ], ℓ = 4 and 6). ✄ This result extends readily to general weights sequences ( η n ) n ≥ .Some technical condi-tions appear on the choice of weights but these conditions are natural and not restrictive(see [Lem05]). In particular we can always consider the choice η n = 1 for which we obtainthe following result: if Γ (2) n √ Γ ( − n n → + ∞ −−−−−→ e β ∈ (0 , + ∞ ), then n q Γ ( − n (cid:0) ν ( ρ ) n ( ω, f ) − ν ( f ) (cid:1) ( R ) = ⇒ N (cid:0) e β m (2) g ; ˆ σ ρ (cid:1) as n → + ∞ . ✄ Polynomial steps.
Let γ n = Cn − µ , µ ∈ (0 , µ > , Γ (3) n → Γ (3) ∞ < + ∞ so that Γ (3) n √ Γ n → n → + ∞ . If µ < , Γ (3) n √ Γ n ≍ n − µ (and if µ = , Γ (3) n √ Γ n ≍ log n √ n ). ConsequentlyΓ (3) n √ Γ n → ⇐⇒ µ > , Γ (3) n √ Γ n → + ∞ ⇐⇒ µ <
15 and Γ (3) n √ Γ n → e β ∈ (0 , + ∞ ) ⇐⇒ µ = 15 . When µ = , e β = C √ √ Γ n ∼ √ C n .As a consequence, if γ n = η n = Cn − , n (cid:0) ν ( ρ ) n ( ω, f ) − ν ( f ) (cid:1) ( R ) = ⇒ N (cid:16) C m (2) g ; 45 b σ ρ C (cid:17) . We switch from a weak rate n to n i.e. a “gain” of n (see figure below). The secondnoticeable fact is that the bias is now significantly more sensitive to the constant C thanin the standard setting. If we minimize the L –norm of the error ν ( ρ ) n ( ω, f ) − ν ( f ) weobtain the optimal choice of C as a function of both bias and standard deviation, precisely C = (cid:18) b σ ρ m (2) q ) (cid:19) . ρ and uniqueness of µ ( I d ) P ROPOSITION
Let ρ be an admissible correlation matrix i.e. such that ρ ∗ ρ ≤ I q .Assume that the duplicated diffusion ( X, X ( ρ ) ) has a unique invariant distribution µ ( ρ ) (sothat if ρ = I q , µ ( I q ) = ν ∆ ). ( a ) b σ ρ ≥ Z R d | σ ∗ ∇ g | dν . ( b ) If ρ = 0 then b σ ρ = 5 Z R d | σ ∗ ∇ g | dν . ( c ) If ρ = I q , b σ ρ = Z R d | σ ∗ ∇ g | dν . O r d e r o f t h e r a t e o f c o n v e r g e n ce Order µ of the step sequence γ n = Cn − µ Crude algorithm RR extrapolation Proof.
Claims ( b ) and ( c ) being obvious thanks to (4.36), we only prove ( a ). Keepingin mind that both marginals µ ( ρ ) ( R d × dy ) and µ ( ρ ) ( dx × R d ) are equal to ν , one derivesthanks to Schwarz’s Inequality (once on R d and once on L ( µ )) from the expression (4.36)of the asymptotic variance b σ ρ that b σ ρ ≥ Z R d | σ ∗ ∇ g | dν − (cid:20)Z R d × R d | σ ∗ ∇ g | ( x ) µ ( ρ ) ( dx, dy ) (cid:21) (cid:20)Z R d × R d | ρσ ∗ ∇ g | ( y ) µ ( ρ ) ( dx, dy ) (cid:21) = 5 Z R d | σ ∗ ∇ g | dν − (cid:20)Z R d | σ ∗ ∇ g | ( x ) ν ( dx ) (cid:21) (cid:20)Z R d | ρσ ∗ ∇ g | ( y ) ν ( dy ) (cid:21) ≥ Z R d | σ ∗ ∇ g | dν − Z R d | σ ∗ ∇ g | dν = Z R d | σ ∗ ∇ g | dν where we used in the last inequality that | ρu | ≤ | u | .The previous result says that the structural asymptotic variance of the RR estimatoris always greater than that of the standard estimator but can be equal if the Brownianmotions are equal. This condition is in fact almost necessary. Actually, thanks to thePythagorean identity, σ ρ = 5 Z R d | σ ∗ ∇ g | dν + 2 Z R d × R d | σ ∗ ∇ g ( x ) − ρσ ∗ ∇ g ( y ) | µ ( ρ ) ( dx, dy ) − Z R d × R d | σ ∗ ∇ g ( x ) | ν ( dx ) − Z R d × R d | ρσ ∗ ∇ g ( y ) | ν ( dy ) . Then, since ρ ∗ ρ ≤ I q , a necessary condition to obtain σ ρ = R R d | σ ∗ ∇ g | dν is | ρσ ∗ ∇ g ( y ) | = | σ ∗ ∇ g ( y ) | ν ( dy )- a.e. When ρ ∗ ρ < I q , this equality can not hold except if σ ∗ ∇ g ( y ) = 0 ν ( dy )- a.e. .3 Proof of Theorem 4.3 Without loss of generality, we assume that f satisfies ν ( f ) = 0 so that f = A g under( C ( f , k )). We denote by γ ( r ) the sequence defined by γ ( r ) k = γ rk . L EMMA
Assume that f satisfies ( C ( f , )) and denote by g the solution to the Poissonequation A g = f . Then, Γ n ¯ ν ( ρ ) n ( ω, f ) = 2 (cid:0) g ( ¯ Y n ) − g ( ¯ Y ) (cid:1) − (cid:0) g ( ¯ X n ) − g ( ¯ X ) (cid:1) − n X k =1 √ γ k (cid:0) √ M (2) k − ∆ M (1) k (cid:1) (4.38) − E n − E n + N n + R n (4.39) where ∆ M (1) k = ( ∇ g ( ¯ X k − ) | σ ( ¯ X k − ) U k ) , ∆ M (2) k = ( ∇ g ( ¯ Y k − ) | σ ( ¯ Y k − ) Z k − ) + ( ∇ g ( ¯ Y k − ) | σ ( ¯ Y k − Z k ) , E n = 2 n X k =1 (cid:16) γ k (cid:17) (cid:0) ϕ ( ¯ Y k − ) + E [ ϕ ( ¯ Y k − ) |F k − ] (cid:1) − n X k =1 γ k ϕ ( ¯ X k − ) , E n = 2 n X k =1 (cid:16) γ k (cid:17) (cid:0) ϕ ( ¯ Y k − ) + E [ ϕ ( ¯ Y k − ) |F k − ] (cid:1) − n X k =1 γ k ϕ ( ¯ X k − ) with ϕ and ϕ defined by (4.32) and (4.37) , ( N n ) is defined by N n = n X k =1 (cid:16) ∆ N ( ¯ Y k − , Z k − , γ k N ( ¯ Y k − , Z k , γ k (cid:17) − ∆ N ( ¯ X k − , Z k − , γ k where ∆ N ( x, U, γ ) = H ( x, U, γ ) − E x [ H ( x, U, γ )] and H ( x, U, γ ) = γ D g ( x )( σ ( x ) U ) ⊗ + 16 X ℓ =0 C − ℓ γ ℓ +32 D g ( x ) b ( x ) ⊗ ℓ ( σ ( x ) U ) ⊗ (3 − ℓ ) + 124 X ℓ =0 γ ℓ +42 C − ℓ D g ( x ) b ( x ) ⊗ ℓ ( σ ( x ) U ) ⊗ (4 − ℓ ) + γ X ℓ =4 C − ℓℓ ℓ ! D ℓ g ( x ) b ( x ) ⊗ (6 − ℓ ) ( σ ( x ) U ) ⊗ ℓ . Finally, if ( S a ) holds, the sequence ( R n ) n ≥ satisfies the following property: there exists r > such that, a.s. , for every n ≥ , E [ | ∆ R n ||F n − ] ≤ Cγ n (cid:0) V r ( ¯ X n − ) + V r ( ¯ Y n − ) + V r ( ¯ Y n − ) (cid:1) (4.40) where ∆ R n = R n − R n − . R EMARK . The above decomposition is built as follows: the second term of (4.38)is the main martingale component of the decomposition whereas E n, contains the firstorder discretization error. Thanks to the Richardson-Romberg extrapolation, E n, is infact negligible when n → + ∞ . When the step sequence decreases fast (Theorem4.3( i )),the rate of convergence is ruled by the main martingale component. In Theorem 4.3( ii ),the rate is ruled by E n, and E n, . Finally, N n contains all the negligible martingale terms.36 roof. Owing to ( C ( f , )), to the Taylor formula and to the fact that E [ D ( x )( σ ( x ) U ) ⊗ ] =Tr( σ ∗ ( x ) D g ( x ) σ ( x )), we have g ( ¯ X k ) = g ( ¯ X k − ) + γ k A g ( ¯ X k − ) + √ γ k ∆ M k, (4.41)+ 12 (cid:0) D ( ¯ X k − )( σ ( ¯ X k − ) U k ) ⊗ − E [ D ( ¯ X k − )( σ ( ¯ X k − ) U k ) ⊗ |F k − ] (cid:1) (4.42)+ X l =3 D l g ( ¯ X k − ) (cid:0) γ k b ( ¯ X k − ) + √ γ k σ ( ¯ X k − ) U k (cid:1) ⊗ l (4.43)+ D g ( ξ k ) (cid:0) γ k b ( ¯ X k − ) + √ γ k σ ( ¯ X k − ) U k (cid:1) ⊗ (4.44)where ξ k ∈ [ ¯ X k − , ¯ X k ]. The fact that |∇ V | ≤ CV implies that √ V is a Lipschitz con-tinuous function with Lipschitz constant denoted by [ √ V ] Lip . Then, setting k D g ( x ) k =sup | α | =7 | ∂ α g ( x ) | and using Assumption ( C ( f , )), we have k D g ( ξ k ) k ≤ C ( √ V ( ξ k )) r ≤ C ( √ V ( ¯ X k − ) + [ √ V ] Lip | ∆ ¯ X k | ) r (4.45)where ∆ ¯ X k = γ k b ( ¯ X k − ) + √ γ k σ ( ¯ X k − ) U k . Then, owing to the elementary inequality | a + b | p ≤ c p ( | a | p + | b | p ) and to Assumption ( S a ), it follows that there exists r > E [ | ∆ R n ||F k − ] ≤ Cγ k V r ( ¯ X k − ) . Then we plug this control into the above Taylor expansion and to compensate the termsof (4.43) when necessary. An appropriate (tedious) grouping of the terms yields: γ k A g ( ¯ X k − ) = g ( ¯ X k ) − g ( ¯ X k − ) − √ γ k ∆ M k, − γ k ϕ ( ¯ X k − ) − γ k ϕ ( ¯ X k − ) − ∆ N ( ¯ X k − , U k , γ k ) − ∆ R n where R n, satisfies (4.40). Making the same development for A g ( ¯ Y k − ) and for A g ( ¯ Y k − )and summing over n yield the announced result. L EMMA
Let a ∈ (0 , such that ( S a ) holds. Assume that ( X t , X ( ρ ) t ) t ≥ admits aunique invariant distribution µ ( ρ ) . Let g be a C -function such that |∇ g | ≤ CV r where r ∈ R + . Then, √ Γ n n X k =1 √ γ k ( √ M (2) k − ∆ M (1) k ) n → + ∞ = ⇒ ˆ σ ρ . Proof.
Let { ξ k,n , k = 1 , . . . , n, n ≥ } be the triangular array of ( F k )-martingale incrementsdefined by ξ k,n = r γ k Γ n ( √ M (2) k − ∆ M (1) k ) . Let us show that n X k =1 E [ | ξ k,n | |F k − ] n → + ∞ −−−−−→ ˆ σ ρ . First, using that Σ U = I q , we obtain that for every k ≥ E [ | ∆ M (1) k | |F k − ] = | σ ∗ ∇ g ( ¯ X k − ) | . x
7→ | σ ∗ ∇ g | ( x ) is a continuous function such that | σ ∗ ∇ g | ≤ CV r for a positive r ,it follows from Proposition (4.9) that1Γ n n X k =1 γ k E [ | ∆ M (1) k | |F k − ] → + ∞ −−−−→ Z R d | σ ∗ ∇ g | ( x ) ν ( dx ) . (4.46)Similarly, E [ | ∆ M (2) k | |F k − ] = | σ ∗ ∇ g ( ¯ Y k − ) | + E [ | σ ∗ ∇ g ( ¯ Y k − ) | |F k − ] . It follows that 1Γ n n X k =1 γ k E [ | ∆ M (2) k | |F k − ] = 2 ν ( ρ ) n ( ω, | σ ∗ ∇ g | ) − n n X k =1 ζ k (4.47)where ( ζ k ) is a sequence of ( F k )-martingale increments defined by ζ k = γ k (cid:0) | σ ∗ ∇ g ( ¯ Y k − ) | − E [ | σ ∗ ∇ g ( ¯ Y k − ) | |F k − ] (cid:1) . Using that | σ ∗ ∇ g | ≤ CV r for a positive real number r , we obtain by similar arguments tothose used in (4.45) that E [ | ζ k | |F k − ] ≤ CV r ( ¯ Y k − ). We derive from Proposition 4.9( i )applied with θ k = k that + ∞ X k =1 E [ (cid:12)(cid:12)(cid:12)(cid:12) ζ k Γ k (cid:12)(cid:12)(cid:12)(cid:12) |F k − ] ≤ Cγ ∞ X k =1 γ k Γ k V r ( ¯ Y k − ) < + ∞ since X k ≥ γ k Γ k ≤ + ∞ X k =2 Z Γ k Γ k − dss ≤ Z + ∞ Γ dss < + ∞ . As a consequence ( P nk =1 ζ k Γ k ) n ≥ is a convergent martingale and the Kronecker Lemmathen implies that n P nk =1 ζ k n → + ∞ −−−−−→ a.s . Thus, we deduce from (4.47) combined withProposition 4.9 that1Γ n n X k =1 γ k E [ | ∆ M (2) k | |F k − ] n → + ∞ −−−−−→ ν ( | σ ∗ ∇ g | ) a.s. (4.48)Finally, we have to manage the cross-product: keeping in mind the construction of thenoises of the Euler schemes (see (4.33) and (4.34), we have: √ E [∆ M (1) k ∆ M k, |F k − ] = (( σ ∗ ∇ g )( ¯ X k − ) | ρ ( σ ∗ ∇ g )( ¯ Y k − ))+ (( σ ∗ ∇ g )( ¯ X k − ) | ρ ( σ ∗ ∇ g )( ¯ Y k − )) − γ − k ζ (2) k where ζ (2) k = γ k (cid:0) (( σ ∗ ∇ g )( ¯ X k − ) | ρ ( σ ∗ ∇ g )( ¯ Y k − )) − E [(( σ ∗ ∇ g )( ¯ X k − ) | ρ ( σ ∗ ∇ g )( ¯ Y k − )) |F k − ] (cid:1) so that 1Γ n n X k =1 √ E [∆ M (1) k ∆ M k, |F k − ] = µ (1) n ( ψ ) + µ (2) n ( ψ ) − n n X k =1 ζ (2) k (4.49)38here ψ : R d → R is defined by ψ ( x, y ) = ( σ ∗ ∇ g ( x ) | ρ ( σ ∗ ∇ g )( y )) and for every Borelfunction f : R d → R , µ (1) n ( f ) = 1Γ n n X k =1 γ k f ( ¯ X k − , ¯ Y k − ) and µ (2) n ( f ) = 1Γ n n X k =1 γ k f ( ¯ X k − , ¯ Y k − ) . By straightforward adaptations of the proof of Proposition 4.9, we can show that if( X t , X ( ρ ) t ) has a unique invariant distribution µ ( ρ ) then, for every continuous function f such that f ≤ CV r with r > µ ( i ) n ( ω, f ) n → + ∞ −−−−−→ µ ( ρ ) ( f ) a.s. with i = 1 , . As a consequence, µ (1) n ( ψ ) + µ (2) n ( ψ ) n → + ∞ −−−−−→ µ ( ψ ) a.s . Finally, by martingale argumentssimilar to those used for ( ζ k ), one checks that Γ − n P nk =1 ζ (2) k n → + ∞ −−−−−→ a.s. Thus, by (4.46),(4.48) and (4.49), we obtain that n X k =1 E [ | ξ k,n | |F k − ] n → + ∞ −−−−−→ ν ( | σ ∗ ∇ g | ) − µ ( ψ ) = ˆ σ ρ . Then, the result follows from the CLT for arrays of martingale increments provided thata Lindeberg-type condition is satisfied (see [HH80], Corollary 3.1). To be precise, it isenough to prove that there exists δ > n X k =1 E [ | ξ k,n | δ |F k − ] n → + ∞ −−−−−→ a.s. (4.50)Using Assumption ( S a ) and the fact |∇ g | ≤ CV r ( r > r > E [ | ξ k,n | δ |F k − ] ≤ C γ δk Γ δn (cid:0) V r ( ¯ X k − ) + V r ( ¯ Y k − ) + V r ( ¯ Y k − ) (cid:1) . Thus, n X k =1 E [ | ξ k,n | δ |F k − ] ≤ C Γ (1+ δ ) n Γ δn (cid:16) ν γ (1+ δ ) n ( V r ) + ν γ (1+ δ ) , ( ρ ) n ( V r ) (cid:17) . Checking easily that Γ (1+ δ ) n Γ δn n → + ∞ −−−−−→
0, (4.50) follows from Proposition 4.9( ii ). L EMMA
Let a ∈ (0 , such that ( S a ) holds. Assume that ( X t ) admits a uniqueinvariant distribution ν . Assume ( C ( f , k )) and that Γ (3) n n → + ∞ −−−−−→ + ∞ . Then,(i) If ϕ defined by (4.32) satisfies ( C ( ϕ , )) then, (3) n E n, → + ∞ −−−−→ − m (1) g ϕ a.s. (ii) If the derivatives of g up to order are continuous and dominated by V r (with r > ), (3) n E n, → + ∞ −−−−→ − ν ( ϕ ) a.s. roof. (i) Writing2 n X k =1 (cid:16) γ k (cid:17) (cid:0) ϕ ( ¯ Y k − ) + E [ ϕ ( ¯ Y k − ) |F k − ] (cid:1) = n X k =1 γ k (cid:0) ϕ ( ¯ Y k − ) + ϕ ( ¯ Y k − ) (cid:1) + n X k =1 γ k T k with ∆ T k being a martingale increment defined by ∆ T k = E [ ϕ ( ¯ Y k − ) |F k − ] − ϕ ( ¯ Y k − ),one obtains that E n, = Γ (2) n h ( ν γ (2) , ( ρ ) n − ν )( ϕ ) − ( ν γ (2) n − ν )( ϕ ) i + n X k =1 γ k T k . Applying Theorem V.3 of [Lem05] (which is an extension of Proposition 4.9( iv ) to generalweights) with η k = γ k and q ∗ = 4, we obtain thatΓ (2) n Γ (3) n ( ν γ (2) n − ν )( ϕ ) n → + ∞ −−−−−→ m g ϕ ∈ R in probability.Similarly, applying this result to the Euler scheme with half-step, we have:Γ (2) n Γ (3) n [( ν γ (2) , ( ρ ) n − ν )( ϕ )] = 12 Γ (2) n P nk =1 γ k . γ k [( ν γ (2) , ( ρ ) n − ν )( ϕ )] n → + ∞ −−−−−→ m g ϕ ∈ R in probability.Thus, it remains to show that the martingale term is negligible. We set θ k = γ k Γ (3)2 k . Usingthat ( γ k ) is non-increasing, one checks that ( θ n ) is non-increasing and that P θ k γ k < + ∞ .Since | ϕ | ≤ CV r with r >
0, it follows from Proposition 4.9 that X k ≥ γ k (Γ (3) k ) E [ | ϕ | ( ¯ Y k − )] < + ∞ . This implies that the martingale P γ k Γ (3) k ∆ T k is a.s. convergent so that the Kronecker lemmayields (3) n P nk =1 γ k ∆ T k n → + ∞ −−−−−→ a.s. . The first assertion follows. (ii) Remark that E n, = 12 ν γ (3) , ( ρ ) n − ν γ (3) n ( ω, ϕ ) + n X k =1 γ k T k . Under the assumptions, ϕ is continuous and dominated by V r with a positive r . Then,since Γ (3) n n → + ∞ −−−−−→ + ∞ , ( ν γ (3) , ( ρ ) n ( ϕ )) n ≥ and ( ν γ (3) n ( ω, ϕ )) n ≥ converge to ν ( ϕ ). Withsome similar arguments as previously, one checks that the martingale term is negligibleand the second assertion follows. For the sake of simplicity, we choose to give the proof of Theorem 4.3 only when Γ (3) n n → + ∞ −−−−−→ + ∞ . Note that if γ n = Cn − µ , this corresponds to µ ≤ / i.e. the case where the Rombergextrapolation really increases the rate of convergence (see Remark 4.9).By the decomposition of Lemma 4.1 and the convergences established in Lemmas 4.2and 4.3, one checks that it is now enough to prove the following points:Θ n Γ n (cid:0) (cid:0) g ( ¯ Y n ) − g ( ¯ Y (cid:1) − ( g ( ¯ X n ) − ¯ X ) (cid:1) P −→ n → + ∞ , (4.51)40 n Γ n N n P −→ n Γ n R n P −→ n → + ∞ (4.52)with Θ n = √ Γ n ∨ Γ n Γ (3) n .For (4.51), the result is obvious when g is bounded. Otherwise, we use Lemma 3 of[LP03] which implies in particular that for every p > E [ V p ( ¯ X n )] ≤ C p Γ n . By Jensen’sinequality, this implies that for every r > α ∈ (0 , C > ∀ n ≥ , E [ V r ( ¯ X n )] ≤ (cid:0) E [ V rα ( ¯ X n )] (cid:1) α ≤ C α rα Γ αn . Thus, since the same property holds for the ( ¯ Y n ) and since | g | ≤ CV r with r >
0, (4.52)follows taking α ∈ (0 , / { π k,n , k =1 , . . . , n, n ≥ } the triangular array of ( F k )-martingale increments defined by π k,n = ∆ N k √ Γ n . Then, in order to prove the convergence in probability of ( N n / √ Γ n ) to 0, we use the CLTfor martingale increments which says that, since a Lindeberg-type condition holds (we donot prove this point, see Proof of Lemma 4.2 for a similar argument), it is enough to showthat n X k =1 E [ | π k,n | |F k − ] n → + ∞ −−−−−→ a.s. (4.53)Under the assumptions on g and on the coefficients, one checks that there exists r > n X k =1 E [ | π k,n | |F k − ] ≤ C n n X k =1 γ k (cid:0) V r ( ¯ X k − ) + V r ( ¯ Y k − ) + V r ( ¯ Y k − ) (cid:1) . By Proposition 4.9, sup n ≥ (cid:16) ν n ( ω, V r ) + ν ( ρ ) n ( ω, V r ) (cid:17) < + ∞ . Assertion (4.53) follows.As concerns R n , it follows from a martingale argument that1Γ (3) n n X k =1 (∆ R k − E [∆ R k |F k − ]) P −→ n → + ∞ . Now, since sup n ≥ (cid:16) ν γ n ( ω, V r ) + ν γ , ( ρ ) n ( ω, V r ) (cid:17) < + ∞ a.s. and since γ n n → + ∞ −−−−−→
0, wededuce that 1Γ (3) n n X k =1 E [∆ R k |F k − ] P −→ . The last assertion follows. 41
Hypo-ellipticity of the correlated duplicated system
It is a well-known fact that, for a Markov process, the strong Feller property combinedwith some irreducibility of the transitions implies uniqueness of the invariant distribution(see e.g. [DPZ96], Theorem 4.2.1). For a diffusion process with smooth coefficients, suchproperties hold if it satisfies the hypoelliptic H¨ormander assumption (see [H¨or67, H¨or85])and if the deterministic system related to the stochastic differential system (written in theStratanovich sense) is controllable. In fact, both properties can be transferred from theoriginal
SDE to the duplicated system so that its invariant distribution is also unique.The main result of this section is Proposition A.11. Before, we need to introduce someH¨ormander-type notations. First, written in a Stratonovich way, X is a solution to dX t = A ( X t ) dt + q X j =1 A j ( X t ) ◦ dW jt (A.54)where A , . . . A q are vectors fields on R d defined by : A ( x ) = d X i =1 b i ( x ) − X l,j σ l,j ( x ) ∂ x j σ i,l ( x ) ∂ x i and for every j ∈ { , . . . , q } : A j ( x ) = d X i =1 σ i,j ( x ) ∂ x i . For the sake of simplicity, we assume that b and σ are C ∞ on R d with bounded derivatives.We will also assume the following H¨ormander condition at each point: there exists N ∈ N ∗ such that ∀ x ∈ R d ,dim (Span { A ( x ) , A ( x ) , . . . , A q ( x ) , L. B. of length ≤ N of the A j ( x )’s , ≤ j ≤ q } ) = d (A.55)where “L.B.” stands for Lie Brackets. The above assumptions imply that for every t > x ∈ R d , P t ( x, . ) admits a density p t ( x, . ) w.r.t. the Lebesgue measure and that ( x, y ) p t ( x, y ) is C ∞ on R d × R d (see e.g. [Cat92], Theorem 2.9). In particular, x P t ( x, . ) is astrong Feller semi-group. Assume also that the control system (associated with (A.54))˙ x ( u ) = A ( x ( u ) ) + q X j =1 A q ( x ( u ) ) u j , (A.56)is approximatively -controllable:There exists T > ε > x , x ∈ R d , there exists u ∈ L ([0 , T ] , R d )such that ( x ( u ) ( t )) solution to (A.56) satisfies x (0) = x and | x ( T ) − x | ≤ ε . (A.57)Under Assumptions (A.55) and (A.57), the diffusion has a unique invariant distribution ν . Actually, the controllability assumption combined with the Support Theorem implies With a standard abuse of notation, we identify the vectors fields and the associated differential oper-ators. O , for every x ∈ R d , P T ( x, O ) >
0. The semi-group( P t ) is then irreducible. Owing to the strong Feller property, it follows classically that ( P t )admits a unique invariant distribution (see e.g. [DPZ96], Proposition 4.1.1. and Theorem4.2.1.).Furthemore, ν is absolutely continuous with respect to the Lebesgue measure on R d and itstopological support is R d (since for every open set O of R d , ν ( O ) = R P T ( x, ν ( dx ) > X t , X ( ρ ) t ). Setting Z ( ρ ) t = ( X t , X ( ρ ) t ) andusing the preceding notations, (1.3) can be written: dZ ( ρ ) t = e A ( Z ( ρ ) t ) dt + q X j =1 e A j ( Z ( ρ ) t ) dW jt + q X j =1 e A d + j ( Z ( ρ ) t ) d f W jt where e A ( z ) = ( A ( x ) , A ( y )) T (with A ( y ) = P di =1 h b i ( y ) − P l,j σ l,j ( y ) ∂ y j σ i,l ( y ) i ∂ y i and z = ( x, y )), f W is a d -dimensional Brownian Motion independent of W such that W ( ρ ) = ρ ∗ W + ( I q − ρ ∗ ρ ) f W and for every j ∈ { , . . . , q } , e A j ( z ) = A j ( x ) + A ( ρ ) j ( y ) and, e A q + j ( z ) = A (( I q − ρ ∗ ρ ) ) j ( y ) (A.58)where for a for a q × q matrix B , A ( B ) j ( y ) = P di =1 ( σ ( y ) B ) i,j ∂ y i . Then, the followingproperty holds. P ROPOSITION
A.11.
Let ρ ∈ M q,q ( R ) such that ρ ∗ ρ < I q . Assume that b and σ are C ∞ on R d with bounded derivatives. Assume (A.55) and (A.57) . Then, uniqueness holds forthe invariant distribution ν ( ρ ) of the duplicated diffusion ( X t , X ( ρ ) t ) . Furthermore, if ν ( ρ ) exists, then ν ( ρ ) has a density p ( ρ ) ( w.r.t. λ d ) which is a.s. positive.Proof. First, let us check the H¨ormander conditions for ( X t , X ( ρ ) t ) t ≥ . Setting S = ( I q − ρ ∗ ρ ) , standard computations yield ∀ j ∈ { , . . . , q } , e A q + j ( z ) = q X l =1 S l,j A l ( y ) . Since S is invertible, we deduce that { A l ( y ) , l = 1 , . . . , q } belongs to Span { e A q + j ( z ) , j =1 , . . . , q } . Similarly, checking that for every j ∈ { , . . . , q } ,[ e A ( z ) , e A q + j ( z )] = [ A ( y ) , A ( S ) j ( y )] = q X l =1 S l,j [ A ( y ) , A l ( y )]one deduces from the invertibility of S that { [ A ( y ) , A l ( y )] , l = 1 , . . . , q } is included inSpan { [ e A ( z ) , e A q + j ( z )] , j = 1 , . . . , q } . Owing to (A.55), it follows that Span { ∂ y , . . . , ∂ y d } is included in V = Span n e A ( z ) , e A ( z ) , . . . , e A q ( z ) , Lie Brackets of length ≤ N of the e A j ( z )’s , ≤ j ≤ q o . Now, let us show that Span { ∂ x , . . . , ∂ x d } is included in V . Since Span { ∂ y , . . . , ∂ y d } isincluded in V , it is clear that for every x ∈ R d , A j ( x ) = A ( ρ ) j ( y ) − e A j ( z ) also belongs to V . Since [ e A ( z ) , e A j ( z )] = [ A ( x ) , A j ( x )] + [ A ( y ) , A ( ρ ) j ( y )] , A ( x ) , A j ( x )] has the same property. Using again (A.55), we deduce that Span { ∂ x , . . . , ∂ x d } is included in V and thus that dim( V ) = 2 d . As a consequence, for every z ∈ R d × R d and t > Q ( ρ ) t ( z, . ) admits a density q t ( z, . ) w.r.t. λ d such that ( z, z ′ ) q t ( z, z ′ ) is C ∞ on R d × R d × R d × R d .In order to obtain uniqueness for the invariant distribution, it remains to show thatthere exists T > z ∈ R d × R d , for every non-empty open set O of R d × R d , Q T ( z, O ) >
0. Owing to (A.57), it is clear that for every z = ( x , y )and z = ( x , y ), for every ε >
0, there exist u and e u ∈ L ([0 , T ] , R d ) such that z ( t ) = ( x ( u ) ( t ) , x ( e u ) ( t )), where x ( u ) and x ( e u ) are solutions to (A.56) starting from x and y , satisfies | z ( T ) − z | ≤ ε . Furthermore, since S is invertible, we can assume that e u = ρu + Sω with ω ∈ L ([0 , T ] , R d ). Then, the support Theorem can be applied to obtainthat for every z , z , ε Q T ( z , B ( z , ε ) > z ∈ R d × R d and every non-empty open set O , Q T ( z, O ) > B Additional proofs about the two-dimensional counterex-ample
Proof of (3.8) : For the sake of completeness, we show that r t → a.s. as soon as r > r t ) defined a.s. by r t = 1 for every t ≥ r = 1. Owing to the strong Markov property, this impliesthat if τ := inf { t ≥ , r t = 1 } , then r t = 1 on { τ ≤ t } . The same property holds at 0.We deduce that ( r t ) t ≥ lives in [1 , + ∞ ) if r > ,
1] if r ∈ [0 , r >
1, we have d ( r t −
1) = − ( r t − dt + ϑdW t ) so that r t − e − (1+ ϑ ) t + ϑW t . It follows that lim t → + ∞ r t = 1 since lim t → + ∞ W t t = 0 a.s. . Now, if r ∈ [0 , dr t = r t (1 − r t )( dt + ϑdW t ) . Thus, ( r t ) is a [0 , r t converges a.s. to a [0 , r ∞ . Since ∀ t ≥ , E [ r t ] = r + E (cid:16) Z t r s (1 − r s ) ds (cid:17) , it follows that E [ R + ∞ r s (1 − r s ) ds ] which in turn implies that R + ∞ r s (1 − r s ) ds < + ∞ a.s. As a consequence lim inf t → + ∞ r t (1 − r t ) = 0 a.s. . The process ( r t ) being a.s. convergentto r ∞ , it follows that r ∞ ∈ { , } a.s. . It remains to prove that P ( r ∞ = 0) = 0. Denoteby p the scale function of ( r t ) null at r = 1 /
2. For every r ∈ (0 , p ( r ) = Z r e − R ξ
12 2 ϑ u (1 − u ) du dξ = Z r (cid:18) − ξξ (cid:19) ϑ dξ. As a consequence, if ϑ ∈ (0 , √ r → + ∞ p ( r ) = + ∞ . This means that 0 is a repulsivepoint and that, as a consequence (see e.g. [KT81], Lemma 6.1 p. 228), ∀ b ∈ (0 , P ( lim a → + τ a < τ b ) := lim a → + P ( τ a < τ b ) = 044here τ a = inf { t ≥ | r t = a } , y ∈ [0 , P ( r ∞ = 0) = 0. This completesthe proof. ✷ Proof of (3.9) : We want to prove that µ is invariant for ( X xt , X x ′ t ) if and only if µ canbe represented by (3.9). First, since the unique invariant distribution of ( X xt ) is λ S , itis clear that µ = L ( e i Θ , e i (Θ + V ) ) where Θ has uniform distribution on [0 , π ] and V isa random variable with values in [0 , π ). One can check that if V is independent of Θ , µ is invariant. Thus, it remains to prove that it is a necessary condition or equivalentlythat K ( θ, dv ) := L ( e iV | e i Θ = e iθ ) does not depend on θ . Denote by ( e i Θ t , e i (Θ t + V t ) ) the(stationary) duplicated diffusion starting from ( e i Θ , e i (Θ + V ) ). Since µ is invariant, wehave for every t ≥ L ( e iV t | e i Θ t = e iθ ) = K ( θ, dv )but thanks to the construction, for every t ≥
0, Θ t = Θ + W t and V t = V (the angulardifference between the two coordinates does not change) so that L ( e iV t | e i Θ t = e iθ ) = Z K ( θ ′ , dv ) ρ t ( θ, dθ ′ )where ρ t ( θ, dθ ′ ) = L ( e i ( θ + W t ) ). But ρ t ( θ, dθ ′ ) converges weakly to λ S when t → + ∞ .From the two previous equations it follows that K ( θ, dv ) does not depend on θ since ∀ θ ≥ K ( θ, dv ) = R K ( θ ′ , dv ) λ S ( dθ ′ ). ✷ Acknowledgement:
We thank an anonymous referee and an associate editor for their suggestionswhich helped improving the paper.
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