Fast flow asymptotics for stochastic incompressible viscous fluids in R 2 and SPDEs on graphs
aa r X i v : . [ m a t h . P R ] S e p Fast flow asymptotics for stochastic incompressible viscous fluidsin R and SPDEs on graphs ∗ Sandra Cerrai † , Mark Freidlin ‡ Department of MathematicsUniversity of MarylandCollege Park, Maryland, USA
Abstract
Fast advection asymptotics for a stochastic reaction-diffusion-advection equation arestudied in this paper. To describe the asymptotics, one should consider a suitable classof SPDEs defined on a graph, corresponding to the stream function of the underlyingincompressible flow.
Consider an incompressible flow in R , with stream function − H ( x ), x ∈ R , and let someparticles move together with the flow. If we denote by u ( t, x ) the density of the particles attime t ≥ x ∈ R , then the function u ( t, x ) satisfies the Liouville equation ∂ t u ( t, x ) = (cid:10) ¯ ∇ H ( x ) , ∇ u ( t, x ) (cid:11) , t > , x ∈ R ,u (0 , x ) = ϕ ( x ) , x ∈ R . (1.1)Suppose now that the flow has a small viscosity and the particles take part in a slow chemicalreaction, with a deterministic and a stochastic component, as described by the equation ∂ t ˜ u ǫ ( t, x ) = ǫ u ǫ ( t, x ) + (cid:10) ¯ ∇ H ( x ) , ∇ ˜ u ǫ ( t, x ) (cid:11) + ǫ b (˜ u ǫ ( t, x )) + √ ǫ g (˜ u ǫ ( t, x )) ∂ t ˜ W ( t, x ) , ˜ u ǫ (0 , x ) = ϕ ( x ) , x ∈ R . (1.2)Here, 0 < ǫ << b, g : R → R are Lipschitz continuous non-linearitiesand the stream function − H : R → R is a generic function, having four continuous derivatives,with bounded second derivative, and such that H ( x ) → + ∞ , as | x | ↑ + ∞ . The noise ˜ W ( t, x ) ∗ Key words : Averaging principle, , Markov processes on graphs, stochastic partial differential equations † Partially supported by the NSF grant DMS 1407615
Asymptotic Problems for SPDEs . ‡ Partially supported by the NSF grant DMS 1411866
Long-term Effects of Small Perturbations and OtherMultiscale Asymptotic Problems .
1s supposed to be a spatially homogeneous Wiener process having finite spectral measure (seeSections 2 and 7 for all assumptions and details).The small positive parameter ǫ is included in equation (1.2) in such a way that all per-turbation terms have strength of the same order, as ǫ ↓
0. It is not difficult to check thatunder the above conditions, if we take the limit as ǫ ↓
0, the solution ˜ u ǫ ( t, x ) of equation (1.2)converges on any finite time interval to the solution u ( t, x ) of equation (1.1), in probability,uniformly with respect to x in a bounded domain of R . But on large time intervals, growingtogether with ǫ − , the difference ˜ u ǫ ( t, x ) − u ( t, x ) can have order 1, as ǫ ↓ u ǫ ( t, x ) =: ˜ u ǫ ( t/ǫ, x ) , t ≥ , x ∈ R . With this change of time, the new function u ǫ ( t, x ) solves the equation ∂ t u ǫ ( t, x ) = 12 ∆ u ǫ ( t, x ) + 1 ǫ (cid:10) ¯ ∇ H ( x ) , ∇ u ǫ ( t, x ) (cid:11) + b ( u ǫ ( t, x )) + g ( u ǫ ( t, x )) ∂ t W ( t, x ) ,u ǫ (0 , x ) = ϕ ( x ) , x ∈ R , (1.3)for some spatially homogeneous Wiener process W ( t, x ).In the present paper, we are interested in the limiting behavior of the solution u ǫ ( t, x ) ofequation (1.3), as ǫ ↓
0, in a finite time interval. In particular, we will see that in order todescribe the limit of u ǫ ( t, x ), one should consider SPDEs on a non standard setting, where thespace variable changes on the graph Γ obtained by identifying all points in each connectedcomponent of the level sets of the Hamiltonian H .A suitable class of SPDEs on a graph has been already studied in our previous paper [1],where SPDEs defined on a net of narrow channels were studied. In that case, we have tried tounderstand what happens of the solution of the SPDE defined on a 2-dimensional channel G with many wings and subject to instantaneous reflections at the boundary, when the width ofthe channel goes to zero. Actually, we have proved that the solution converges to the solutionof a suitable SPDE, defined on a suitable graph that can be associated with the channel, in L p (Ω; C ([ τ, T ]; L ( G ))), for any 0 < τ < T .Here we are considering the case of a reaction-diffusion-advection equation in R , wherethe reaction term has a deterministic and a stochastic component, and the advection term isof order ǫ − , compared to the diffusion and the reaction part. For every fixed ǫ >
0, the secondorder differential operator L ǫ defined by L ǫ ϕ ( x ) = 12 ∆ ϕ ( x ) + 1 ǫ (cid:10) ¯ ∇ H ( x ) , ∇ ϕ ( x ) (cid:11) , x ∈ R , is associated with the stochastic equation dX ǫ ( t ) = 1 ǫ ¯ ∇ H ( X ǫ ( t )) dt + dw ( t ) , X ǫ (0) = x ∈ R , (1.4)for some 2-dimensional Brownian motion w ( t ), defined on a stochastic basis (Ω , F , {F t } t ≥ , P ).This means that u ǫ is a mild solution to equation (1.3) if it satisfies u ǫ ( t ) = S ǫ ( t ) ϕ + Z t S ǫ ( t − s ) B ( u ǫ ( s )) ds + Z t S ǫ ( t − s ) G ( u ǫ ( s )) d W ( s ) , B and G are the composition/multiplication functionals associated with b and g , re-spectively, (see Section 7 for the definition), and S ǫ ( t ) is the Markov transition semigroupassociated with equation (1.4) S ǫ ( t ) ϕ ( x ) = E x ϕ ( X ǫ ( t )) , t ≥ , x ∈ R . In [4, Chapter 8] it is proved that, if Π is the projection of R onto the graph Γ, thenfor any x ∈ R and T > X ǫ ( · )) converges, in the sense of weak convergenceof distributions in C ([0 , T ]; Γ), to a Markov process ¯ Y on Γ. Namely, for every continuousfunctional F defined on C ([0 , T ]; Γ) and any x ∈ R it holdslim ǫ → E x F (Π( X ǫ ( · ))) = ¯ E Π( x ) F ( ¯ Y ( · )) . (1.5)The generator ¯ L of the process ¯ Y is explicitly given, in terms of suitable second order differentialoperators defined on each edge of the graph and suitable gluing conditions at the vertices.As a consequence of the limiting result (1.5), in [4, Chapter 8] Freidlin and Wentcell havealso studied the asymptotic behavior of the solution of the elliptic problem
12 ∆ f ǫ ( x ) + 1 ǫ (cid:10) ¯ ∇ H ( x ) , ∇ f ǫ ( x ) (cid:11) = − g, x ∈ D,f ǫ ( x ) = ρ ( x ) , x ∈ ∂D, where D is a bounded smooth domain in R and g and ρ are continuous functions on D and ∂D ,respectively. Actually, they have proven that f ǫ converges to the solution of the correspondingelliptic equation on the graph, associated with the operator ¯ L . In [5], Ishii and Souganidis,by using only deterministic arguments, have proved an analogous result in the more generalsituation the Laplace operator is replaced with the operator Tr[ A ( x ) D ], where A is a smooth,symmetric, non-negative matrix-valued mapping defined on D Next, in [3] the limiting behavior of the solution of the deterministic parabolic problem ∂ t v ǫ ( t, x ) = 12 ∆ v ǫ ( t, x ) + 1 ǫ (cid:10) ¯ ∇ H ( x ) , ∇ v ǫ ( t, x ) (cid:11) + b ( v ǫ ( t, x )) ,v ǫ (0 , x ) = ϕ ( x ) , x ∈ R , (1.6)has been studied. Under the crucial assumption that the projection of the support of thefunction ϕ on the graph Γ does not contain any vertex, it is shown that for any 0 < τ < T lim ǫ → sup t ∈ [ τ,T ] | u ǫ ( t, x ) − ¯ v ( t, Π( x )) | = 0 , (1.7)uniformly with respect to x from any compact set of R , where ¯ v is the solution of the parabolicproblem on Γ ∂ t ¯ v ( t, z, k ) = ¯ L ¯ v ( t, z, k ) + b (¯ v ( t, z, k )) , ¯ v (0 , z, k ) = ϕ ∧ ( z, k ) := 1 T k ( z ) I C k ( z ) ϕ ( x ) |∇ H ( x ) | dl z,k . (1.8)3ere dl z,k is the surface measure on the connected component C k ( z ) of the level set C ( z ) = { x ∈ R : H ( x ) = z } , corresponding to the edge I k , and T k ( z ) = I C k ( z ) |∇ H ( x ) | dl z,k , (see Section 2 for all details).Assuming that the projection of the support of the initial condition ϕ on the graph Γdoes not contain any vertex, allows to avoid to deal with the vertices points, where seriousdiscontinuity problems arise. Actually, in order to prove (1.11), it is necessary to prove thatfor any ϕ ∈ C b ( R ) lim ǫ → sup t ∈ [ τ,T ] (cid:12)(cid:12) E x (cid:2) ϕ ( X ǫ ( t )) − ϕ ∧ (Π( X ǫ ( t ))) (cid:3)(cid:12)(cid:12) = 0 , (1.9)and lim ǫ → sup t ∈ [ τ,T ] (cid:12)(cid:12) E x ϕ ∧ (Π( X ǫ ( t ))) − ¯ E Π( x ) ϕ ∧ ( ¯ Y ( t )) (cid:12)(cid:12) = 0 . (1.10)Limit (1.9) has been obtained in [3], as a consequence of the averaging principle, by using angle-action coordinates, away from the vertices. Limit (1.10) was obtained in [3] as an immediateconsequence of (1.5), since the function ϕ ∧ is continuous away from the vertices.But the assumption that the support of the initial condition on the graph Γ does notcontain any vertex it too restrictive and it is critical in several situations to be able to prove(1.11) for a general ϕ ∈ C b ( R ). In particular, this is necessary when dealing with SPDEs, asin this case we cannot assume that the support of the noise satisfies such a condition.For this reason, in Section 5 we prove that for any general function ϕ ∈ C b ( R )lim ǫ → sup t ∈ [ τ,T ] (cid:12)(cid:12) S ǫ ( t ) u ( x ) − ( ¯ S ( t ) u ∧ ) ◦ Π( x ) (cid:12)(cid:12) = 0 , (1.11)with x ∈ R and 0 < τ < T fixed. Also in this case, (1.11) follows once we prove limits (1.9)and (1.10), but their proofs are considerably more delicate than in [3], due to the presence ofvertices. Actually, in order to prove (1.9) and (1.10) we have to consider separately the regionof R close to the critical points of the Hamiltonian and the region far from them and introducesuitable sequences of stopping times that allow to go from one region to the other. By usingthe fact that the process X ǫ spends a small amount of time close to the critical points, weobtain suitable nice properties of those stopping times that allow us to conclude the validityof (1.11) for general functions ϕ .Next, we go back to the SPDE (1.3), where, as we mentioned above, W ( t, x ) is a spacehomogeneous Wiener process, having finite spectral measure µ . This means, that we canrepresent W ( t, x ) as W ( t, x ) = ∞ X j =1 d u j µ ( x ) β j ( t ) , t ≥ , for some complete orthonormal system { u j } j ∈ N in L ( R ; dµ ) and a sequence of independentBrownian motions { β j } j ∈ N , all defined on the same stochastic basis.We study equation (1.3) in a space of square integrable functions, with respect to a weightedmeasure γ ◦ Π( x ) dx on R . In fact, the choice of the weight γ is not trivial, as we have to choose4t in such a way that γ ◦ Π is admissible with respect to all semigroups S ǫ ( t ), its projection γ onthe graph Γ is admissible with respect to the semigroup ¯ S ( t ) and functions on L ( R , γ ◦ Π dx )are projected to functions in L ( γ, dν γ ), where ν γ is the projection of the Lebesgue measureon Γ, with weight γ . Moreover, we need to show that from (1.11) we obtain the limitlim ǫ → sup t ∈ [ τ,T ] (cid:12)(cid:12) S ǫ ( t ) u − ( ¯ S ( t ) u ∧ ) ◦ Π (cid:12)(cid:12) L ( R ,γ ◦ Π( x ) dx ) = 0 . (1.12)Once identified the right class of weights, we introduce the process¯ W ( t, z, k ) = ∞ X j =1 ( d u j µ ) ∧ ( z, k ) β j ( t ) , t ≥ z, k ) ∈ Γ , and we show that ¯ W ( t ) ∈ L ( Ω ; L (Γ , dν γ )), for every t ≥
0. In particular, as we can provethat k ¯ S ( t ) k L ( L (Γ ,dν γ )) ≤ c T , t ∈ [0 , T ] , and g is Lipschitz continuous, we obtain that for any v ∈ L p ( Ω ; C ([0 , T ]; L (Γ , dν γ ))) thestochastic convolution t ∈ [0 , + ∞ ) Z t ¯ S ( t − s ) G ( v ( s )) d ¯ W ( s ) = ∞ X k =1 Z t ¯ S ( t − s ) (cid:2) G ( v ( s ))( d u j µ ) ∧ (cid:3) dβ j ( s ) , is well defined in L ( Ω ; C ([0 , T ]; L (Γ , dν γ ))) and the mapping u Z · ¯ S ( · − s ) G ( v ( s )) d ¯ W ( s )is Lipschitz continuous in L ( Ω ; C ([0 , T ]; L (Γ , dν γ ))).In particular, as also b is Lipschitz continuous, this implies that the SPDE ∂ t ¯ u ( t, z, k ) = ¯ L ¯ u ( t, z, k ) + b (¯ u ( t, z, k )) + g (¯ u ( t, z, k )) ∂ t ¯ W ( t, z, k ) , t > , ¯ u (0 , z, k ) = ϕ ∧ ( z, k ) , ( z, k ) ∈ Γ . is well posed in L ( Ω ; C ([0 , T ]; L (Γ , dν γ ))). Finally, as a consequence of (1.12), we can provethe main result of this paper, namelylim ǫ → E sup t ∈ [ τ,T ] | u ǫ ( t ) − ¯ u ( t ) ◦ Π | pL ( R ,γ ◦ Π( x ) dx ) = 0 . To conclude, we would like to stress the fact that the techniques we are currently developingto deal with SPDEs on graphs can be further developed to treat more sophisticated andcomplex situations. For example, one could consider the case there are several different typesof particles, instead of only one, as in the present paper. In this case we expect to get asystem of SPDEs on a graph. Moreover, if the original flow does not moves on R , but on a2-dimensional surface (for example a 2 D torus), with a positive genus, the underling dynamicscan have delays at some vertices. This leads to certain effects for SPDEs on the graph thatare worth of investigation. Finally, in the multidimensional case, when several conservationlaws are present, we expect will obtain SPDEs on a generalization of a graph, namely an openbook (see [4, Chapter 9]). We are planning to address in forthcoming papers.5 Notations and preliminaries
We consider here the Hamiltonian system˙ X ( t ) = ¯ ∇ H ( X ( t )) , X (0) = x ∈ R . (2.1)Throughout the present paper, we will assume that the Hamiltonian H is a generic function,with non degenerate critical points, and having quadratic growth, for | x | → ∞ . More precisely Hypothesis 1.
The mapping H : R → R satisfies the following conditions.1. It is four times continuously differentiable, with bounded second derivative.2. It has only a finite number of critical points x , . . . , x n . The matrix of second derivatives D H ( x i ) is non degenerate, for every i = 1 , . . . , n and H ( x i ) = H ( x j ) , if i = j .3. There exist three positive constants a , a , a such that H ( x ) ≥ a | x | , |∇ H ( x ) | ≥ a | x | and ∆ H ( x ) ≥ a , for all x ∈ R , with | x | large enough.4. We have min x ∈ R H ( x ) = 0 . Notice that we can always assume condition 4. without any loss of generality.Once introduced the Hamiltonian H , for every z ≥
0, we denote by C ( z ) the z -level set C ( z ) = (cid:8) x ∈ R : H ( x ) = z (cid:9) . The set C ( z ) may consist of several connected components C ( z ) = N ( z ) [ k =1 C k ( z ) , and for every x ∈ R we have X (0) = x = ⇒ X ( t ) ∈ C k ( x ) ( H ( x )) , t ≥ , where C k ( x ) ( x ) is the connected component of the level set C ( H ( x )), to which the point x belongs. For every z ≥ k = 1 , . . . , N ( z ), we shall denote by G k ( z ) the domain of R bounded by the level set component C k ( z ). Γ and the identification map Π If we identify all points in R belonging to the same connected component of a given level set C ( z ) of the Hamiltonian H , we obtain a graph Γ, given by several intervals I , . . . I n and vertices O , . . . , O m . The vertices will be of two different types, external and internal vertices. Externalvertices correspond to local extrema of H , while internal vertices correspond to saddle pointsof H . Among external vertices, we will also include O ∞ , the endpoint of the only unboundedinterval in the graph, corresponding to the point at infinity.6n what follows, we shall denote by Π : R → Γ the identification map , that associatesto every point x ∈ R the corresponding point Π( x ) on the graph Γ. We have Π( x ) =( H ( x ) , k ( x )), where k ( x ) denotes the number of the interval on the graph Γ, containing thepoint Π( x ). If O i is one of the interior vertices, the second coordinate cannot be chosen in aunique way, as there are three edges having O i as their endpoint. Notice that both k ( x ) and H ( x ) are first integrals (a discrete and a continuous one, respectively) for the Hamiltoniansystem (2.1).On the graph Γ, a distance can be introduced in the following way. If y = ( z , k ) and y = ( z , k ) belong to the same edge I k , then d ( y , y ) = | z − z | . In the case y and y belongto different edges, then d ( y , y ) = min (cid:8) d ( y , O i ) + d ( O i , O i ) + · · · + d ( O i j , y ) (cid:9) , where the minimum is taken over all possible paths from y to y , through every possiblesequence of vertices O i , . . . , O i j , connecting y to y . If z is not a critical value, then each C k ( z ) consists of one periodic trajectory of the vectorfield ¯ ∇ H ( x ). If z is a local extremum of H ( x ), then, among the components of C ( z ) there isa set consisting of one point, the rest point of the flow. If H ( x ) has a saddle point at somepoint x and H ( x ) = z , then C ( z ) consists of three trajectories, the equilibrium point x andthe two trajectories that have x as their limiting point, as t → ±∞ .We introduce here some other notations related to the Hamiltonian H , that will be usedthroughout the paper.- For every edge I k , we denote G k = n x ∈ R : Π( x ) ∈ ˚ I k o . - For every 0 ≤ z < z and for every edge I k , we denote G k ( z , z ) = G k ( z , z ) = { x ∈ G k : z < H ( x ) < z } . - For every δ >
0, we denote G ( ± δ ) = m [ i =1 G i ( ± δ ) = m [ i =1 (cid:8) x ∈ R : H ( O i ) − δ < H ( x ) < H ( O i ) + δ (cid:9) , (here, with some abuse of notation, we denote by H ( O i ) the value of the Hamiltonian H at those x ∈ R such that Π( x ) = O i ).- For every vertex O i and edge I k , we denote D i = (cid:8) x ∈ R : Π( x ) = O i (cid:9) , D ik = D i ∩ ¯ G k . z ≥ H on the set ¯ G k , we denote C k ( z ) = (cid:8) x ∈ ¯ G k : H ( x ) = z (cid:9) = C ( z ) ∩ ¯ G k . - For every δ > I k and vertex O i such that I k ∼ O i , we denote D ( ± δ ) = [ k,i : I k ∼ O i D ik ( ± δ ) = [ k,i : I k ∼ O i { x ∈ G k : dist(Π( x ) , O i ) = δ } . Now, for every ( z, k ) ∈ Γ, we define T k ( z ) = I C k ( z ) |∇ H ( x ) | dl z,k , α k ( z ) = I C k ( z ) |∇ H ( x ) | dl z,k , (2.2)where dl z,k is the length element on C k ( z ). Notice that T k ( z ) is the period of the motion alongthe level set C k ( z ).As we have seen above, if X (0) = x ∈ C k ( z ), then X ( t ) ∈ C k ( z ), for every t ≥
0. Asknown, for every ( z, k ) ∈ Γ the probability measure dµ z,k := 1 T k ( z ) 1 |∇ H ( x ) | dl z,k (2.3)is invariant for system (2.1) on the level set C k ( z ). Moreover, it is possible to prove that forany 0 ≤ z < z and k = 1 , . . . n Z G k ( z ,z ) u ( x ) dx = Z I k ∩ [( z ,k ) , ( z ,k )] I C k ( z ) u ( x ) |∇ H ( x ) | dl z,k dz. (2.4)In particular, if we take u ≡
1, we getarea( G k ( z , z )) = Z I k ∩ [( z ,k ) , ( z ,k )] T k ( z ) dz, so that T k ( z ) = dS k ( z ) dz , (2.5)where S k ( z ) is the area of the domain G k ( z ) bounded by the level set C k ( z ). Finally, by thedivergence theorem, it is immediate to check that α k ( z ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z G k ( z ) ∆ H ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z G k ( z ) ω H ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , where ω H is the vorticity of the flow.As discussed in [4, section 8.1], since the Hamiltonian H has only non-degenerate criticalpoints, if ( z, k ) approaches the endpoint of an edge I k , corresponding to an external vertex O i = ( H ( x j ) , k ), then lim ( z,k ) → O i T k ( z ) = 2 π p det [Hess H ( x j )] > . (2.6)8ig.1 1: The Hamiltonian, the level sets, the projection and the graphIf ( z, k ) approaches the endpoint of an edge I k , corresponding to an internal vertex O i =( H ( x j ) , k ), then T k ( z ) = I C k ( z ) |∇ H ( x ) | dl z,k ∼ const | log | z − H ( x i ) || → + ∞ , as ( z, k ) → O i . (2.7)Finally, if ( z, k ) approaches O ∞ , we have T k ( z ) = O (1) , as ( z, k ) → O ∞ . (2.8)Actually, for | x | large, we have H ( x ) ≥ a | x | , so that S k ( z ) = O ( z ) and hence, due to (2.5),(2.8) follows.Next, we would like to recall that in [4, Lemma 8.1.1] it is proven that if v ∈ C ( R ), thenfor every fixed k the mapping z I C k ( z ) v ( x ) |∇ H ( x ) | dl z,k , is continuously differentiable with respect to z such that ( z, k ) ∈ ˚ I k , and ddz I C k ( z ) v ( x ) |∇ H ( x ) | dl z,k = I C k ( z ) h∇ v ( x ) , ∇ H ( x ) i + v ( x )∆ H ( x ) |∇ H ( x ) | dl z,k . (2.9) We consider here the following random perturbation of system (2.1) d ˜ X ǫ ( t ) = ¯ ∇ H ( ˜ X ǫ ( t )) dt + √ ǫ d ˜ w ( t ) , ˜ X ǫ (0) = x, (2.10)9here ˜ w ( t ) is a two dimensional Wiener process and ǫ > t ∈ [0 , + ∞ ) H ( ˜ X ǫ ( t )) ∈ R is not constant any more. Actually,the motion ˜ X ǫ ( t ) consists of a fast rotation along the deterministic unperturbed trajectoriesand a slow motion across them.In what follows, it will be convenient to do a change of time and, with the time rescaling t t/ǫ , the process ˜ X ǫ ( t/ǫ ) will coincide in distribution with the solution of the stochasticequation dX ǫ ( t ) = 1 ǫ ¯ ∇ H ( X ǫ ( t )) dt + dw ( t ) , X ǫ (0) = x, (2.11)where w ( t ) is another two dimensional Wiener process. We will denote by S ǫ ( t ), t ≥
0, theMarkov transition semigroup associated with equation (2.11). Namely, for every Borel boundedfunction ϕ : R → R S ǫ ( t ) ϕ ( x ) = E x ϕ ( X ǫ ( t )) , t ≥ , x ∈ R . Now, for every x ∈ R , we consider the process Π( X ǫ ( t )), t ≥
0, defined on the graphΓ, with X ǫ (0) = x . In [4, Chapter 8] it has been studied the limiting behavior, as ǫ ↓
0, ofthe process Π( X ǫ ) in the space C ([0 , T ]; Γ), for any fixed T > x ∈ R . Namely, in [4,Theorem 8.2.2] it has been proved that the process Π( X ǫ ), which describes the slow motionof the motion X ǫ , converges, in the sense of weak convergence of distributions in the space ofcontinuous Γ-valued functions, to a diffusion process ¯ Y on Γ.The process ¯ Y has been described in [4, Theorem 8.2.1] in terms of its generator ¯ L , whichis given by suitable differential operators ¯ L k within each edge I k of the graph and by certaingluing conditions at the interior vertices O i of the graph.For each k = 1 , . . . , n , the differential operator ¯ L k , acting on functions f defined on theedge I k , has the form ¯ L k f ( z ) = 12 T k ( z ) ddz (cid:18) α k dfdz (cid:19) ( z ) , (2.12)where T k ( z ) and α k ( z ) are the mappings defined in (2.2). The operator ¯ L , acting on functions f defined on the graph Γ, is defined as¯ Lf ( z, k ) = ¯ L k f ( z ) , if ( z, k ) is an interior point of the edge I k . It is immediate to check that each operator ¯ L k can be represented as ¯ L k = ( d/dv k )( d/du k ),where v ′ k ( z ) = I C k ( z ) |∇ H ( x ) | dl z,k , u ′ k ( z ) = I C k ( z ) |∇ H ( x ) | dl z,k ! − . In view of this representation, by studying the limiting behavior of the function u k and v k atthe vertices of the graph Γ, it is possible to check that the internal vertices, correspondingto the saddle points of H , are accessible, while the external vertices and the vertex O ∞ areinaccessible.The domain D ( ¯ L ) is defined as the set of continuous functions on the graph Γ, that aretwice continuously differentiable in the interior part of each edge of the graph, such that forany vertex O i = ( z i , k i ) = ( z i , k i ) = ( z i , k i ) there exist finitelim ( z,k ij ) → O i ¯ Lf ( z, k i j ) , j = 1 , , , I k ij ∼ O i . Moreover, for each interior vertex O i thefollowing gluing condition is satisfied X j =1 ± α k ij ( z i ) d k ij f ( z i , k i j ) = 0 , (2.13)where d k ij is the differentiation along I k ij and the sign + is taken if the H -coordinate increasesalong I k ij and the sign − is taken otherwise.The operator ( ¯ L, D ( ¯ L ) is a non-standard operator, because it is a differential operator ona graph, endowed with suitable gluing conditions and because it is degenerate at the verticesof the graph. Nevertheless in [4, Theore 8.2.1] it is shown that it is the generator of a Markovprocess ¯ Y on the graph Γ. In what follows, we shall denote by ¯ P ( z,k ) and ¯ E ( z,k ) the probabilityand the expectation associated with ¯ Y , starting from ( z, k ) ∈ Γ. Moreover, we shall denoteby ¯ S ( t ) the semigroup associated with ¯ Y , defined by¯ S ( t ) f ( z, k ) = ¯ E ( z,k ) f ( ¯ Y ( t )) , for every bounded Borel function f : Γ → R .As we mentioned above, in [4, Theorem 8.2.2] it has been proved that the process Π( X ǫ ( · ))is weakly convergent to ¯ Y in C ([0 , T ]; Γ), for every T > x ∈ R . Namely, for everycontinuous functional F defined on C ([0 , T ]; Γ) and x ∈ R lim ǫ → E x F (Π( X ǫ ( · ))) = ¯ E Π( x ) F ( ¯ Y ( · )) . (2.14) Γ We fix here a continuous mapping γ : Γ → (0 , + ∞ ) such that n X k =1 Z I k γ ( z, k ) T k ( z ) dz < ∞ , (3.1)where, we recall, T k ( z ) = I C k ( z ) |∇ H ( x ) | dl z,k . If we define γ ∨ ( x ) = γ (Π( x )) , x ∈ R , we have that γ ∨ : R → (0 , + ∞ ) is a bounded continuous function in L ( R ). Actually, due to(2.4) we have Z R γ ∨ ( x ) dx = n X k =1 Z I k I C k ( z γ ∨ ( x ) |∇ H ( x ) | dl z,k dz = n X k =1 Z I k γ ( z, k ) I C k ( z |∇ H ( x ) | dl z,k dz = n X k =1 Z I k γ ( z, k ) T k ( z ) dz < ∞ .
11n what follows, we shall define H γ = (cid:26) u : R → R : Z R | u ( x ) | γ ∨ ( x ) dx < ∞ (cid:27) = L ( R , γ ∨ ( x ) dx ) , and ¯ H γ = ( f : Γ → R : n X k =1 Z I k | f ( z, k ) | γ ( z, k ) T k ( z ) dz < ∞ ) = L (Γ , ν γ ) , where the measure ν γ is defined as ν γ ( A ) := n X k =1 Z I k I A ( z, k ) γ ( z, k ) T k ( z ) dz, A ⊆ B (Γ) . Now, for every u : R → R we define u ∧ ( z, k ) = 1 T k ( z )) I C k ( z ) u ( x ) |∇ H ( x ) | dl z,k = I C k ( z ) u ( x ) dµ z,k , ( z, k ) ∈ Γ , and for every f : Γ → R we define f ∨ ( x ) = f (Π( x )) , x ∈ R . Proposition 3.1.
Assume the Hamiltonian H satisfies Hypothesis 1 and γ : Γ → (0 , + ∞ ) isa weight function satisfying condition (3.1) . Then, the following holds.1. For every u ∈ H γ we have u ∧ ∈ ¯ H γ and | u ∧ | ¯ H γ ≤ | u | H γ . (3.2)
2. For every f ∈ ¯ H γ we have f ∨ ∈ H γ and | f ∨ | H γ = | f | ¯ H γ . (3.3) Proof.
Let u ∈ H γ . Recalling how the probability measure µ z,k has been defined in (2.3), dueto (2.4) we have | u ∧ ( x ) | H γ = n X k =1 Z I k | u ∧ ( z, k ) | γ ( z, k ) T k ( z ) dz = n X k =1 Z I k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)I C k ( z ) u ( x ) dµ z,k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) γ ( z, k ) T k ( z ) dz ≤ n X k =1 Z I k I C k ( z ) | u ( x ) | dµ z,k γ ( z, k ) T k ( z ) dz = n X k =1 Z I k I C k ( z ) | u ( x ) | γ ∨ ( x ) |∇ H ( x ) | dl z,k dz = Z R | u ( x ) | γ ∨ ( x ) dx = | u | H γ . This implies that u ∧ ∈ ¯ H γ and (3.2) holds. 12oncerning the second part of the Proposition, by using again (2.4) for every f ∈ ¯ H γ wehave | f ∨ | H γ = Z R | f ∨ ( x ) | γ ∨ ( x ) dx = Z R | f (Π( x )) | γ (Π( x )) dx = n X k =1 Z I k I C k ( z ) | f ( z, k ) | γ ( z, k ) |∇ H ( x ) | dl z,k dz = n X k =1 Z I k | f ( z, k ) | γ ( z, k ) T k ( z ) dz = | f | H γ . This allows to conclude that f ∨ ∈ H γ and (3.3) holds.For every u ∈ H γ and f ∈ ¯ H γ , we have (cid:10) f, u ∧ (cid:11) ¯ H γ = (cid:10) f ∨ , u (cid:11) H γ . (3.4)Actually, we have (cid:10) f, u ∧ (cid:11) ¯ H γ = n X k =1 Z I k f ( z, k ) u ∧ ( z, k ) γ ( z, k ) T k ( z ) dz = n X k =1 Z I k f ( z, k ) I C k ( z ) u ( x ) dµ z,k γ ( z, k ) T k ( z ) dz = n X k =1 Z I k I C k ( z ) f (Π( x )) u ( x ) γ ∨ ( x ) |∇ H ( x ) | dl z,k dz, so that, thanks to (2.4), we can conclude (cid:10) f, u ∧ (cid:11) ¯ H γ = Z R f (Π( x )) u ( x ) γ ∨ ( x ) dx = (cid:10) f ∨ , u (cid:11) H γ . Moreover, for every f ∈ ¯ H γ and u ∈ H γ , we have (cid:0) f ∨ u (cid:1) ∧ ( z, k ) = I C k ( z ) f ∨ ( x ) u ( x ) dµ z,k = I C k ( z ) f (Π( x )) u ( x ) dµ z,k = f ( z, k ) I C k ( z ) u ( x ) dµ z,k = f ( z, k ) u ∧ ( z, k ) , so that (cid:0) f ∨ u (cid:1) ∧ = f u ∧ . (3.5)As a consequence of (3.4) and (3.5), we conclude that for any f, g ∈ ¯ H γ h f, g i ¯ H γ = h (cid:0) f ∨ (cid:1) ∧ , g i ¯ H γ = h f ∨ , g ∨ i H γ . In particular, if { f n } n ∈ N is an orthonormal system in ¯ H γ , it follows that the system { ( f n ) ∨ } n ∈ N is orthonormal in H γ . 13 emma 3.2. Assume u ∈ C ( R ) . Then, under Hypothesis 1, the mapping u ∧ is continuouslydifferentiable with respect to z in the set S k ˚ I k . Moreover, for every fixed δ > and M > ( z,k ) ∈ S h [ ˚ I h ∩ Π( G ( ± δ )) c ] z ≤ M (cid:12)(cid:12)(cid:12)(cid:12) ∂u ∧ ∂z ( z, k ) (cid:12)(cid:12)(cid:12)(cid:12) =: c δ,M < ∞ . (3.6) Proof. If u ∈ C ( R ), then the mapping v ( x ) = u ( x ) |∇ H ( x ) | , x ∈ [ k ˚ I k , is continuously differentiable. As we have seen in Subsection 2.2, due to [4, Lemma 8.1.1], thisimplies that the mapping z I C k ( z ) u ( x ) |∇ H ( x ) | dl z,k = I C k ( z ) v ( x ) |∇ H ( x ) | dl z,k is continuously differentiable for every z such that ( z, k ) ∈ ˚ I k . Moreover, thanks to (2.9),for every δ > z such that ( z, k ) ∈ Π( G ( ± δ )) c ∩ ˚ I k . In particular, if we take u = 1, we get that the mapping z T k ( z ) iscontinuously differentiable and the derivative is uniformly bounded with respect to z such that( z, k ) ∈ Π( G ( ± δ )) c ∩ ˚ I k . Therefore, as u ∧ ( z, k ) = I C k ( z ) u ( x ) dµ z,k = 1 T k ( z ) I C k ( z ) u ( x ) |∇ H ( x ) | dl z,k , and T k ( z ) remains uniformly bounded from zero for ( z, k ) ∈ Π( G ( ± δ )) c ∩ ˚ I k , we can conclude.For every Q ∈ L ( H γ ) and f ∈ ¯ H γ , we define Q ∧ f = ( Qf ∨ ) ∧ . Thanks to Proposition 3.1, we have that k Q ∧ k L ( ¯ H γ ) ≤ k Q k L ( H γ ) . In an analogous way, for any A ∈ L ( ¯ H γ ) and u ∈ H γ , we define A ∨ u = ( Au ∧ ) ∨ . As above, due to Proposition 3.1, we have that A ∨ ∈ L ( H γ ) and k A ∨ k L ( H γ ) ≤ k A k L ( ¯ H γ ) . Moreover, due to (3.5), we can check immediately that ( A ∨ ) ∧ = A .14 The semigroup S ǫ ( t ) in the weighted space H γ Since div ¯ ∇ H = 0, it is immediate to check that the Lebesgue measure is invariant for thesemigroup S ǫ ( t ), for every fixed ǫ >
0. In particular, S ǫ ( t ) can be extended to a contractionsemigroup on L p ( R p ), for every p ≥ ǫ > S ǫ ( t ) in the weighted space H γ Hypothesis 2.
The semigroup S ǫ ( t ) is well defined on H γ , for every ǫ > . Moreover, forevery fixed T > , there exists c T > such that k S ǫ ( t ) k L ( H γ ) ≤ c T , t ∈ [0 , T ] , ǫ > . (4.1)In next proposition we will show that, in fact, Hypothesis 2 is fulfilled in some relevantcases. Proposition 4.1.
Assume that the Hamiltonian H satisfies Hypothesis 1. Then, there existsa strictly positive continuous function γ : Γ → (0 , + ∞ ) , satisfying (3.1) , such that Hypothesis2 is verified.Proof. According to condition 3. in Hypothesis 1, we can fix z > max i =1 ,...,n H ( x i )such that H ( x ) ≥ z = ⇒ H ( x ) ≥ a | x | . Once fixed z , we take a positive decreasing function h ∈ C ([0 , + ∞ )), such that h ( t ) = ( exp( − λ (cid:0) √ t − √ z (cid:1) ) , t ≥ z , , t ≤ z , for some arbitrary λ >
0, and define γ ( z, k ) = h ( z ) , ( z, k ) ∈ Γ . In view of (2.6) and (2.7), it is immediate to check that for every interval I k not having O ∞ as its end point, Z I k T k ( z ) dz < ∞ , so that, as | h ( t ) | ≤
1, we get Z I k γ ( z, k ) T k ( z ) dz < ∞ . (4.2)Moreover, if I k ∼ O ∞ , due to (2.8), since h ( t ) has exponential decay, we have that (4.2) holdsas well and hence we can conclude that (3.1) holds.Now, recalling that γ ∨ ( x ) = γ (Π( x )) , we have that γ ∨ ( x ) = h ( H ( x )) , x ∈ R . h ∈ C ([0 , + ∞ )), this implies that γ ∨ ∈ C ( R ). Moreover,∆ γ ∨ ( x ) = h ′′ ( H ( x )) |∇ H ( x ) | + h ′ ( H ( x ))∆ h ( x ) , x ∈ R . Hence, as for any t ≥ z we have h ′ ( t ) = − λ h ( t ) t − , h ′′ ( t ) = λ h ( t ) (cid:16) λt − + t − (cid:17) , we obtain ∆ γ ∨ ( x ) = λ γ ∨ ( x ) " |∇ H ( x ) | H ( x ) λ + 1 H ( x ) ! − H ( x ) H ( x ) . Since |∇ H ( x ) | ≤ c | x | and | ∆ H ( x ) | ≤ c , for every x ∈ R , and H ( x ) ≥ a | x | , if H ( x ) ≥ z ,we have sup x ∈ R ,H ( x ) ≥ z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) |∇ H ( x ) | H ( x ) λ + 1 H ( x ) ! − H ( x ) H ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) =: κ λ < ∞ , so that H ( x ) ≥ z = ⇒ ∆ γ ∨ ( x ) ≤ λ κ λ γ ∨ ( x ) . (4.3)On the other hand, since γ ∨ ∈ C ( R ), we have thatsup x ∈ R ,H ( x ) < z | ∆ γ ∨ ( x ) | =: M < ∞ , and hence H ( x ) < z = ⇒ ∆ γ ∨ ( x ) ≤ M ≤ h ( H ( x )) M = 2 γ ∨ ( x ) M. Together with (4.3), this implies that there exists some c > γ ∨ ( x ) ≤ c γ ∨ ( x ) , x ∈ R . (4.4)The generator of the semigroup S ǫ ( t ) is the second order differential operator L ǫ = 12 ∆ + 1 ǫ ¯ ∇ H ( x ) · ∇ . As div ¯ ∇ H ( x ) = 0, it is immediate to check that the adjoint L ⋆ǫ of the operator L ǫ in L ( R )is given by L ⋆ǫ = 12 ∆ − ǫ ¯ ∇ H ( x ) · ∇ . This implies that the adjoint semigroup S ⋆ǫ ( t ) is the Markov transition semigroup associatedwith the equation dY ǫ ( t ) = − ǫ ¯ ∇ H ( Y ǫ ( t )) dt + d ˆ w ( t ) , Y ǫ (0) = x, w ( t ). Now, by using the Itˆo formula, we have γ ∨ ( Y ǫ ( t )) = γ ∨ ( x ) − ǫ Z t h∇ γ ∨ ( Y ǫ ( s )) , ¯ ∇ H ( Y ǫ ( s )) i ds + Z t h∇ γ ∨ ( Y ǫ ( s )) , dw ( s ) i + 12 Z t ∆ γ ∨ ( Y ǫ ( s )) ds, and then, since ∇ γ ∨ ( x ) = h ′ ( H ( x )) ∇ H ( x ) , x ∈ R , it follows γ ∨ ( Y ǫ ( t )) = γ ∨ ( x ) + Z t h∇ γ ∨ ( Y ǫ ( s )) , dw ( s ) i + 12 Z t ∆ γ ∨ ( Y ǫ ( s )) ds. In view of (4.4), by taking the expectation of both sides above, we get S ⋆ǫ ( t ) γ ∨ ( x ) = ˆ E x γ ∨ ( Y ǫ ( t )) ≤ γ ∨ ( x ) + c Z t ˆ E x γ ∨ ( Y ǫ ( s )) ds = γ ∨ ( x ) + c Z t S ⋆ǫ ( s ) γ ∨ ( x ) ds, and, thanks to the Gronwall lemma, this allows to conclude that S ⋆ǫ ( t ) γ ∨ ( x ) ≤ e c t γ ∨ ( x ) , x ∈ R , t ≥ . (4.5)Now, for any u ∈ H γ , we have | S ǫ ( t ) u | H γ = Z R | S ǫ ( t ) u ( x ) | γ ∨ ( x ) dx ≤ Z R S ǫ ( t ) | u | ( x ) γ ∨ ( x ) dx = h S ǫ ( t ) | u | , γ ∨ i L ( R ) = h| u | , S ⋆ǫ ( t ) γ ∨ i L ( R ) . Then, from (4.5), we have | S ǫ ( t ) u | H γ ≤ e c T h| u | , γ ∨ i L ( R ) = e c T | u | H γ , t ∈ [0 , T ] , which implies (4.1), with c T = e cT . For every ǫ >
0, we consider the linear parabolic Cauchy problem associated with the secondorder differential operator L ǫ ∂ t v ǫ ( t, x ) = L ǫ v ǫ ( t, x ) , t > , x ∈ R ,v ǫ (0 , x ) = g ( x ) , x ∈ R . (5.1)17he solution of problem (5.1) has a probabilistic representation in terms of the Markovtransition semigroup S ǫ ( t ) associated with equation (2.11), that, we recall, is defined for anybounded Borel function ϕ : R → R by S ǫ ( t ) ϕ ( x ) = E x ϕ ( X ǫ ( t )) , x ∈ R . Actually, as a consequence of Itˆo’s formula, if the initial condition ϕ is taken in C b ( R ), then ddt [ S ǫ ( · ) ϕ ( x )]( t ) = L ǫ ( S ǫ ( t ) ϕ ) ( x ) , t ≥ , x ∈ R . (5.2)Moreover, as the Hamiltonian H is assumed to be of class C ( R ), with bounded second deriva-tive, we have that the semigroup S ǫ ( t ) has a smoothing effect, namely it maps Borel boundedfunctions into C b ( R ), for any t >
0. Thanks to the semigroup law, this allows to concludethat (5.2) is satisfied on R , for any Borel bounded function ϕ and for all t > H , in addition to all conditions inHypothesis 1, satisfies also the following condition. Hypothesis 3.
It holds dT k ( z ) dz = 0 , ( z, k ) ∈ Γ . (5.3) Remark 5.1.
Condition (5.3) rules out the case H ( x ) = | x | . This means that with ourmethod we cannot treat the case ¯ ∇ H ( x ) is linear in order to prove the main result statedbelow in (5.12).Our purpose here is to study the asymptotic behavior of S ǫ ( t ) ϕ ( x ), and hence of v ǫ ( t, x ),as ǫ ↓
0. As a consequence of (2.14), if the Hamiltonian H satisfies Hypothesis 1, then for anycontinuous mapping ψ : Γ → R we havelim ǫ → S ǫ ( t ) ψ ∨ ( x ) = ¯ S ( t ) ψ (Π( x )) , t ≥ , x ∈ R . (5.4)As a first thing, we are going to prove that the limit above is uniform with respect to t ∈ [0 , T ],for every fixed T >
0. To this purpose, we introduce some notation.For any η > z η ≥ max i =1 ,...,n H ( x i ) + 1 , (5.5)(we recall x , . . . , x n are the critical points of the Hamiltonian H ), such that, for all ǫ > P x ( ρ ǫ,η ≤ T ) ≤ η, ¯ P Π( x ) ( ρ η ≤ T ) ≤ η, (5.6)where ρ ǫ,η := inf { t ≥ H ( X xǫ ( t )) ≥ z η } , ρ η := inf { t ≥ ¯ Y ( t ) ≥ z η } , (5.7)(here we have denoted π ( z, k ) = z ). Actually, as proved in [4, Lemma 3.2] the family { Π( X ǫ ( · )) } ǫ ∈ (0 ,ǫ ) is tight in C ([0 , T ]; R ). This means that there exists some z η > P x sup t ≤ T H ( X ǫ ( t )) ≥ z η ! ≤ η, and (5.6) follows. 18 roposition 5.2. For every
T > and ψ ∈ C b (Γ) , we have lim ǫ → sup t ∈ [0 ,T ] (cid:12)(cid:12) E x ψ ∨ ( X ǫ ( t )) − ¯ E Π( x ) ψ ( ¯ Y ( t )) (cid:12)(cid:12) = 0 . (5.8) Proof.
For every ǫ >
0, let us define f ǫ ( t ) := E x ψ ∨ ( X ǫ ( t )) − ¯ E Π( x ) ψ ( ¯ Y ( t )) . For every fixed t ≥
0, we have lim ǫ → f ǫ ( t ) = 0 . (5.9)If we prove that the family of functions { f ǫ } ǫ> is equibounded and equicontinuous in C ([0 , T ]),by the Ascoli-Arzel`a Theorem we have that the limit in (5.9) is uniform.Now, the equiboundedness of { f ǫ } ǫ> follows from the fact that ψ is bounded. In order toprove the equicontinuity of { f ǫ } ǫ> , first of all we notice that we may assume that both ψ and ψ ∨ are uniformly continuous. Actually, due to (5.6) for every η > (cid:12)(cid:12) E x ψ ∨ ( X ǫ ( t )) − ¯ E Π( x ) ψ ( ¯ Y ( t )) (cid:12)(cid:12) ≤ (cid:12)(cid:12) E x (cid:0) ψ ∨ ( X ǫ ( t )) ; ρ ǫ,η ≤ T (cid:1)(cid:12)(cid:12) + (cid:12)(cid:12) ¯ E Π( x ) (cid:0) ψ ( ¯ Y ( t )) ; ρ η ≤ T (cid:1)(cid:12)(cid:12) + (cid:12)(cid:12) E x (cid:0) ψ ∨ ( X ǫ ( t )) ; ρ ǫ,η > T (cid:1) − ¯ E Π( x ) (cid:0) ψ ( ¯ Y ( t )) ; ρ η > T (cid:1)(cid:12)(cid:12) ≤ k ψ k ∞ η + (cid:12)(cid:12) E x (cid:0) ψ ∨ ( X ǫ ( t )) ; ρ ǫ,η > T (cid:1) − ¯ E Π( x ) (cid:0) ψ ( ¯ Y ( t )) ; ρ η > T (cid:1)(cid:12)(cid:12) . Therefore, as ψ ∨ is uniformly continuous on { H ( x ) ≤ z η } and ψ is uniformly continuous on { z ≤ z η } , due to the arbitrariness of η we can conclude.Now, if ψ are uniformly continuous, for every η > δ η > | Π( x ) − Π( x ) < δ η , dist (( z , k ) , ( z , k )) < δ η = ⇒ | ψ ∨ ( x ) − ψ ∨ ( x ) | + | ψ ( z , k ) − ψ ( z , k ) | < η . (5.10)Moreover, as { Π( X ǫ ) } ǫ> ⊂ C ([0 , T ]; Γ) is tight, there exists a compact set K η ⊂ C ([0 , T ]; Γ),such that P x (cid:0) Π( X ǫ ) ∈ K cη (cid:1) + ¯ P Π( x ) (cid:0) ¯ Y ∈ K cη (cid:1) ≤ η k ψ k ∞ . (5.11)Since functions in K η are equicontinuous, there exists θ η > K η | t − s | < θ η = ⇒ | Π( X ǫ ( t )) − Π( X ǫ ( s )) | < δ η , | ¯ Y ( t ) − ¯ Y ( s ) | < δ η . Therefore, thanks to (5.10) and (5.11), for every t, s ∈ [0 , T ] such that | t − s | < θ η | f ǫ ( t ) − f ǫ ( s ) |≤ (cid:12)(cid:12) E x (cid:0) ψ ∨ ( X ǫ ( t )) − ψ ∨ ( X ǫ ( s )) ; Π( X ǫ ) ∈ K η (cid:1)(cid:12)(cid:12) + (cid:12)(cid:12) ¯ E Π( x ) (cid:0) ψ ( ¯ Y ( t )) − ψ ( ¯ Y ( s )) ; ¯ Y ∈ K η (cid:1)(cid:12)(cid:12) +2 k ψ k ∞ (cid:0) P x (cid:0) Π( X ǫ ) ∈ K cη (cid:1) + ¯ P Π( x ) (cid:0) ¯ Y ∈ K cη (cid:1)(cid:1) ≤ η k ψ k ∞ η k ψ k ∞ = η.
19n what follows, we want to show that, in fact, under suitable conditions, limit (5.8) is alsotrue if ψ ∨ is replaced by u : R → R and ψ is replaced by u ∧ . Namely, we want to prove thefollowing result. Theorem 5.3.
Assume that the Hamiltonian H satisfies all conditions in Hypotheses 1 and3. Then, for any u ∈ C b ( R ) and x ∈ R , and for any < τ ≤ T , we have lim ǫ → sup t ∈ [ τ,T ] (cid:12)(cid:12) S ǫ ( t ) u ( x ) − ¯ S ( t ) ∨ u ( x ) (cid:12)(cid:12) = lim ǫ → sup t ∈ [ τ,T ] (cid:12)(cid:12) E x u ( X ǫ ( t )) − ¯ E Π( x ) u ∧ ( ¯ Y ( t )) (cid:12)(cid:12) = 0 . (5.12) Before proceeding with the proof of Theorem 5.3 and of all required preliminary results, weintroduce some notations. For every ǫ, η > < δ ′ < δ , by using the notations introducedin Subsection 2.2 we define σ ǫ,η,δ,δ ′ n = min n t ≥ τ ǫ,η,δ,δ ′ n : X ǫ ( t ) ∈ G ( ± δ ) c o ,τ ǫ,η,δ,δ ′ n = min n t ≥ σ ǫ,η,δ,δ ′ n − : X ǫ ( t ) ∈ D ( ± δ ′ ) ∪ C ( z η ) o , (5.13)with τ ǫ,η,δ,δ ′ = 0 and z η defined as in (5.5). Clearly, after the process X ǫ ( t ) reaches C ( z η ), all τ ǫ,η,δ,δ ′ n and σ ǫ,η,δ,δ ′ n coincide with the stopping time ρ ǫ,η introduced in (5.7), and for any n ∈ N X ǫ ( σ ǫ,η,δ,δ ′ n ) ∈ D ( ± δ ) ∪ C ( z η ) , X ǫ ( τ ǫ,η,δ,δ ′ n ) ∈ D ( ± δ ′ ) ∪ C ( z η ) . (5.14)Moreover, if X ǫ (0) ∈ G ( ± δ ) c , we have that σ ǫ,η,δ,δ ′ = 0 and τ ǫ,η,δ,δ ′ is the first time the process X ǫ touches D ( ± δ ′ ) ∪ C ( z η ). In particular, if X ǫ (0) ≥ z η , then τ ǫ,η,δ,δ ′ is the first time theprocess X ǫ touches C ( z η ) and all successive stopping times coincide with ρ ǫ,η . Lemma 5.4.
Assume that the same assumptions of Theorem 5.3 are verified. Then, for every u ∈ C b ( R ) and x ∈ R , and for every < τ < T lim ǫ → sup t ∈ [ τ,T ] (cid:12)(cid:12) E x ( u ∧ ) ∨ ( X ǫ ( t )) − ¯ E Π( x ) u ∧ ( ¯ Y ( t )) (cid:12)(cid:12) = 0 . (5.15) Proof.
Thanks to (5.8), if u ∧ were a continuous function on Γ, then (5.15) would follow imme-diately. Unfortunately, because of the presence of the interior vertices, even if u is continuouson R we cannot conclude that u ∧ is continuous on Γ, in general. This means that we have totreat separately the internal vertices and the rest of the points of the graph Γ.First of all, we fix δ > f δ ∈ C b (Γ) such that k f δ k ∞ ≤ k u ∧ k ∞ ≤ k u k ∞ , f δ = u ∧ , on Π( G ( ± δ/ c ) . Thus, for every δ ∈ (0 ,
1) and t ≥
0, we can write E x ( u ∧ ) ∨ ( X ǫ ( t )) − ¯ E Π( x ) u ∧ ( ¯ Y ( t )) = E x (cid:2) ( u ∧ ) ∨ ( X ǫ ( t )) − f ∨ δ ( X ǫ ( t )) (cid:3) + (cid:2) E x f ∨ δ ( X ǫ ( t )) − ¯ E Π( x ) f δ ( ¯ Y ( t )) (cid:3) + ¯ E Π( x ) (cid:2) f δ ( ¯ Y ( t )) − u ∧ ( ¯ Y ( t )) (cid:3) =: I ǫ,δ ( t ) + I ǫ,δ ( t ) + I δ ( t ) .
20f we prove that for any η > δ η , ǫ η > t ∈ [ τ,T ] (cid:16) | I ǫ,δ η ( t ) | + | I δ η ( t ) | (cid:17) < η, ǫ < ǫ η , (5.16)then sup t ∈ [ τ,T ] (cid:12)(cid:12) E x ( u ∧ ) ∨ ( X ǫ ( t )) − ¯ E Π( x ) u ∧ ( ¯ Y ( t )) (cid:12)(cid:12) ≤ η + sup t ∈ [ τ,T ] (cid:12)(cid:12)(cid:12) E x f ∨ δ η ( X ǫ ( t )) − ¯ E Π( x ) f δ η ( ¯ Y ( t )) (cid:12)(cid:12)(cid:12) , ǫ < ǫ η . Since f δ η ∈ C b (Γ), due to (5.8) and the arbitrariness of η >
0, this implies (5.15).Thus, let us prove (5.16). If we take η ′ = η/ k u k ∞ , due to (5.6) we have I ǫ,δ ( t )= E x (cid:0) ( u ∧ ) ∨ ( X ǫ ( t )) − f ∨ δ ( X ǫ ( t )) ; t < ρ ǫ,η ′ (cid:1) + E x (cid:0) ( u ∧ ) ∨ ( X ǫ ( t )) − f ∨ δ ( X ǫ ( t )) ; t ≥ ρ ǫ,η ′ (cid:1) ≤ E x (cid:0) ( u ∧ ) ∨ ( X ǫ ( t )) − f ∨ δ ( X ǫ ( t )) ; t < ρ ǫ,η ′ (cid:1) + 2 k u k ∞ P x ( t ≥ ρ ǫ,η ′ ) ≤ E x (cid:0) ( u ∧ ) ∨ ( X ǫ ( t )) − f ∨ δ ( X ǫ ( t )) ; t < ρ ǫ,η ′ (cid:1) + η J ǫ,η ′ ,δ ( t ) + η . (5.17)Recalling that f δ = u ∧ on Π( G ( ± δ/ c , we have that f ∨ δ = ( u ∧ ) ∨ on G ( ± δ/ c , so that J ǫ,η ′ ,δ ( t ) = E x ( u ∧ ) ∨ ( X ǫ ( t )) − f ∨ δ ( X ǫ ( t )) ; t ∈ ∞ [ n =0 h τ ǫ,η ′ ,δ,δ/ n , σ ǫ,η ′ ,δ,δ/ n (cid:17)! = ∞ X n =0 E x (cid:16) ( u ∧ ) ∨ ( X ǫ ( t )) − f ∨ δ ( X ǫ ( t )) ; t ∈ h τ ǫ,η ′ ,δ,δ/ n , σ ǫ,η ′ ,δ,δ/ n (cid:17)(cid:17) =: ∞ X n =0 J ǫ,η ′ ,δ ,n ( t ) . (5.18)Due to the strong Markov property, we have (cid:12)(cid:12)(cid:12) J ǫ,η ′ ,δ ,n ( t ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) E x (cid:16)(cid:2) ( u ∧ ) ∨ ( X ǫ ( t )) − f ∨ δ ( X ǫ ( t )) (cid:3) I { t ≥ τ ǫ,η ′ ,δ,δ/ n } I { σ ǫ,η ′ ,δ,δ/ n >t } (cid:17)(cid:12)(cid:12)(cid:12) ≤ E x (cid:16) I { τ ǫ,η ′ ,δ,δ/ n ≤ t } (cid:12)(cid:12)(cid:12) E X ǫ ( τ ǫ,η ′ ,δ,δ/ n ) (cid:16) ( u ∧ ) ∨ ( X ǫ ( t )) − f ∨ δ ( X ǫ ( t )) ; t < σ ǫ,η ′ ,δ,δ/ (cid:17)(cid:12)(cid:12)(cid:12)(cid:17) . Thanks to (5.14), this implies | J ǫ,η ′ ,δ ,n ( t ) |≤ P x (cid:16) τ ǫ,η ′ ,δ,δ/ n ≤ t (cid:17) sup y ∈ D ( ± δ/ ∪ C ( z ′ η ) (cid:12)(cid:12)(cid:12) E y (cid:16) ( u ∧ ) ∨ ( X ǫ ( t )) − f ∨ δ ( X ǫ ( t )) ; t < σ ǫ,η ′ ,δ,δ/ (cid:17)(cid:12)(cid:12)(cid:12) ≤ e t E x e − τ ǫ,η ′ ,δ,δ/ n sup y ∈ D ( ± δ/ ∪ C ( z η ′ ) (cid:12)(cid:12)(cid:12) E y (cid:16) ( u ∧ ) ∨ ( X ǫ ( t )) − f ∨ δ ( X ǫ ( t )) ; t < σ ǫ,η ′ ,δ,δ/ (cid:17)(cid:12)(cid:12)(cid:12) . (5.19)21ccording to what proved in [4, Section 8.3, see (8.3.14)], there exist a constant c > δ > x ∈ R and for all δ ≤ δ and ǫ > ∞ X n =0 E x e − τ ǫ,η ′ ,δ,δ/ n = ∞ X n =0 E x (cid:18) e − τ ǫ,η ′ ,δ,δ/ n ; τ ǫ,η ′ ,δ,δ/ n ≤ ρ ǫ,η ′ (cid:19) ≤ cδ . (5.20)Therefore, from (5.19) we get ∞ X n =0 | J ǫ,η ′ ,δ ,n ( t ) | ≤ cδ k u k ∞ e t t sup y ∈ D ( ± δ/ E y σ ǫ,η ′ ,δ,δ/ . (5.21)Because of our definition, σ ǫ,η ′ ,δ,δ/ is the first exit time of the process X ǫ ( t ) from G ( ± δ ). In[4, Section 8.5, (8.5.17)] it is proved that there exists δ > δ < δ and ǫ > E y σ ǫ,η ′ ,δ,δ/ ≤ c δ | log δ | , y ∈ G ( ± δ ) . (5.22)In particular, due to (5.17), (5.18) and (5.21), this implies that for all δ < δ := δ ∧ δ and all ǫ small enough sup t ∈ [ τ,T ] | I ǫ,δ ( t ) | ≤ cδ k u k ∞ e T τ δ | log δ | + η . This means that for any η > ǫ ,η > δ ,η > t ∈ [ τ,T ] | I ǫ,δ ( t ) | ≤ η , ǫ < ǫ ,η , δ < δ ,η . (5.23)Concerning I δ ( t ), recalling how f δ was defined, for every δ > | I δ ( t ) | ≤ k u k ∞ ¯ P Π( x ) (cid:0) ¯ Y ( t ) ∈ Π( G ( ± δ/ (cid:1) ≤ k u k ∞ ¯ E Π( x ) ψ δ ( ¯ Y ( t )) ≤ k u k ∞ (cid:12)(cid:12) ¯ E Π( x ) ψ δ ( ¯ Y ( t )) − E x ψ ∨ δ ( X ǫ ( t )) (cid:12)(cid:12) + 2 k u k ∞ E x ψ ∨ δ ( X ǫ ( t )) , (5.24)where ψ δ is a function in C b (Γ) such that I Π( G ( ± δ/ ≤ ψ δ ≤ , ψ δ ≡ , on Π( G ( ± ( δ )) c . Now, by proceeding as in the proof of (5.23), we can find ǫ ,η > δ ,η > k u k ∞ sup t ∈ [ τ,T ] E x ψ ∨ δ ( X ǫ ( t )) ≤ η , ǫ ≤ ǫ ,η , δ ≤ δ ,η . (5.25)Therefore, if we set δ η := δ ,η ∧ δ ,η , from (5.23) and (5.24) we getsup t ∈ [ τ,T ] (cid:16) | I ǫ,δ η ( t ) | + | I δ η ( t ) | (cid:17) ≤ η k u k ∞ sup t ∈ [ τ,T ] (cid:12)(cid:12)(cid:12) ¯ E Π( x ) ψ δ η ( ¯ Y ( t )) − E x ψ ∨ δ η ( X ǫ ( t )) (cid:12)(cid:12)(cid:12) , for every ǫ ≤ ǫ ,η ∧ ǫ ,η . As ψ δ η ∈ C b (Γ), due to (5.8), this implies that there exists ǫ η ≤ ǫ ,η ∧ ǫ ,η such that (5.16) holds and hence (5.15) follows.22 .2 Proof of Theorem 5.3 In Lemma 5.4 we have proved that for any u ∈ C b ( R ) and x ∈ R and for any 0 < τ < T lim ǫ → sup t ∈ [ τ,T ] (cid:12)(cid:12) E x ( u ∧ ) ∨ ( X ǫ ( t )) − ¯ E Π( x ) u ∧ ( ¯ Y ( t )) (cid:12)(cid:12) = 0 . Thus, in order to prove (5.12), it is sufficient to prove thatlim ǫ → sup t ∈ [ τ,T ] (cid:12)(cid:12) E x (cid:2) u ( X ǫ ( t )) − ( u ∧ ) ∨ ( X ǫ ( t )) (cid:3)(cid:12)(cid:12) = 0 . (5.26)In what follows, we shall assume that u ∈ C b ( R ). Actually, if this is not the case, we canfix a sequence { u n } n ∈ N ⊂ C b ( R ) such thatlim n →∞ k u − u n k ∞ = 0 , k u n k ∞ ≤ k u k ∞ . Since this also implies that lim n →∞ k ( u ∧ ) ∨ − ( u ∧ n ) ∨ k ∞ = 0 , we get lim n →∞ sup ǫ> sup t ∈ [ τ,T ] | E x [ u ( X ǫ ( t )) − u n ( X ǫ ( t ))( X ǫ ( t ))] | = lim n →∞ sup ǫ> sup t ∈ [ τ,T ] (cid:12)(cid:12) E x (cid:2) ( u ∧ n ) ∨ ( X ǫ ( t )) − ( u ∧ ) ∨ ( X ǫ ( t )) (cid:3)(cid:12)(cid:12) = 0 . Therefore, in order to prove (5.26), we have to prove that for any fixed n ∈ N lim ǫ → sup t ∈ [ τ,T ] (cid:12)(cid:12) E x (cid:2) u n ( X ǫ ( t )) − ( u ∧ n ) ∨ ( X ǫ ( t )) (cid:3)(cid:12)(cid:12) = 0 . To this purpose, let us fix α > ǫ > τ − ǫ α >
0. If we fix η > η ′ = η/ k u k ∞ , we havesup t ≥ (cid:12)(cid:12) E x (cid:0) u ( X ǫ ( t )) − ( u ∧ ) ∨ ( X ǫ ( t )) ; t − ǫ α ≥ ρ ǫ,η ′ (cid:1)(cid:12)(cid:12) ≤ η , ǫ > , where ρ ǫ,η ′ is the stopping time defined in (5.7) and satisfying (5.6). This implies (cid:12)(cid:12) E x (cid:0) u ( X ǫ ( t )) − ( u ∧ ) ∨ ( X ǫ ( t )) (cid:1)(cid:12)(cid:12) ≤ (cid:12)(cid:12) E x (cid:0) u ( X ǫ ( t )) − ( u ∧ ) ∨ ( X ǫ ( t )) ; t − ǫ α < ρ ǫ,η ′ (cid:1)(cid:12)(cid:12) + η . (5.27)Now, as in the proof of Lemma 5.4, we have E x (cid:0) u ( X ǫ ( t )) − ( u ∧ ) ∨ ( X ǫ ( t )) ; t − ǫ α < ρ ǫ,η ′ (cid:1) = X n ∈ N E x (cid:16) u ( X ǫ ( t )) − ( u ∧ ) ∨ ( X ǫ ( t )) ; t − ǫ α ∈ [ τ ǫ,η ′ ,δ,δ/ n , σ ǫ,η ′ ,δ,δ/ n ) (cid:17) + X n ∈ N E x (cid:16) u ( X ǫ ( t )) − ( u ∧ ) ∨ ( X ǫ ( t )) ; t − ǫ α ∈ [ σ ǫ,η ′ ,δ,δ/ n , τ ǫ,η ′ ,δ,δ/ n +1 ) (cid:17) =: X n ∈ N J ǫ,η ′ ,δ ,n ( t ) + X n ∈ N J ǫ,η ′ ,δ ,n ( t ) . (5.28)23s in the proof of Lemma 5.4, due to (5.21), we have that there exist δ > c > ǫ sufficiently small and δ < δ X n ∈ N J ǫ,η ′ ,δ ,n ( t ) ≤ cδ k u k ∞ e t − ǫ α t − ǫ α sup y ∈ D ( ± δ/ E y σ ǫ,η ′ ,δ,δ/ , so that, thanks to (5.22), we getsup t ∈ [ τ,T ] X n ∈ N J ǫ,η ′ ,δ ,n ( t ) ≤ c k u k ∞ e T τ − ǫ α δ | log δ | . (5.29)On the other hand, by using once more the strong Markov property, we have | J ǫ,η ′ ,δ ,n ( t ) |≤ E x (cid:16) I { σ ǫ,η ′ ,δ,δ/ n ≤ t − ǫ α } (cid:12)(cid:12)(cid:12) E X ǫ ( σ ǫ,η ′ ,δ,δ/ n ) (cid:16) u ( X ǫ ( t )) − ( u ∧ ) ∨ ( X ǫ ( t )) ; t − ǫ α < τ ǫ,η ′ ,δ,δ/ (cid:17)(cid:12)(cid:12)(cid:12)(cid:17) ≤ P x (cid:16) σ ǫ,η ′ ,δ,δ/ n ≤ t − ǫ α (cid:17) sup y ∈ D ( ± δ ) (cid:12)(cid:12)(cid:12) E y (cid:16) u ( X ǫ ( t )) − ( u ∧ ) ∨ ( X ǫ ( t )) ; t − ǫ α < τ ǫ,η ′ ,δ,δ/ (cid:17)(cid:12)(cid:12)(cid:12) ≤ e t − ǫ α E x (cid:18) e − τ ǫ,η ′ ,δ,δ/ n (cid:19) sup y ∈ D ( ± δ ) (cid:12)(cid:12)(cid:12) E y (cid:16) u ( X ǫ ( t )) − ( u ∧ ) ∨ ( X ǫ ( t )) ; t − ǫ α < τ ǫ,η ′ ,δ,δ/ (cid:17)(cid:12)(cid:12)(cid:12) , and thanks to (5.20), this implies that there exists δ >
0, such that for any δ ≤ δ ∞ X n =1 | J ǫ,η ′ ,δ ,n ( t ) | ≤ c e t − ǫ α δ sup y ∈ D ( ± δ ) (cid:12)(cid:12)(cid:12) E y (cid:16) u ( X ǫ ( t )) − ( u ∧ ) ∨ ( X ǫ ( t )) ; t − ǫ α < τ ǫ,η ′ ,δ,δ/ (cid:17)(cid:12)(cid:12)(cid:12) . For every y ∈ R , we have E y (cid:16) u ( X ǫ ( t )) − ( u ∧ ) ∨ ( X ǫ ( t )) ; t − ǫ α < τ ǫ,η ′ ,δ,δ/ (cid:17) = E y (cid:16) u ( X ǫ ( t )) − ( u ∧ ) ∨ ( X ǫ ( t − ǫ α )) ; t − ǫ α < τ ǫ,η ′ ,δ,δ/ (cid:17) + E y (cid:16) ( u ∧ ) ∨ ( X ǫ ( t − ǫ α )) − ( u ∧ ) ∨ ( X ǫ ( t )) ; t − ǫ α < τ ǫ,η ′ ,δ,δ/ , t < τ ǫ,η ′ ,δ,δ/ (cid:17) + E y (cid:16) ( u ∧ ) ∨ ( X ǫ ( t − ǫ α )) − ( u ∧ ) ∨ ( X ǫ ( t )) ; t − ǫ α < τ ǫ,η ′ ,δ,δ/ , t ≥ τ ǫ,η ′ ,δ,δ/ (cid:17) =: X i =1 L ǫ,δi ( t, y ) . Let us start considering L ǫ,δ ( t ). If y ∈ D ( ± δ ), then τ ǫ,η ′ ,δ,δ/ is the first time the process X ǫ touches D ( ± δ/ ∪ C ( z η ′ ). This means that t < τ ǫ,η ′ ,δ,δ/ = ⇒ X ǫ ( s ) ∈ G ( ± δ/ c ∩ { H ≤ z η ′ } , s ≤ t.
24n particular, Π( X ǫ ( s )) remains in the interior of the same edge of the graph Γ where y is, forall s ≤ t < τ ǫ,η ′ ,δ,δ/ . As we are assuming that u ∈ C b ( R ), due to Lemma 3.2 we have that u ∧ is continuously differentiable on Π( G ( ± δ/ c , with uniformly bounded derivative.As a consequence of Itˆo’s formula, for every s < t we have H ( X ǫ ( t )) − H ( X ǫ ( s )) = 12 Z ts ∆ H ( X ǫ ( r )) dr + Z ts h∇ H ( X ǫ ( r )) , dw ( r ) i . Hence, since for y ∈ D ( ± δ ) and s < t < τ ǫ,η ′ ,δ,δ/ , the process Π( X ǫ ( s )) remains in the sameedge of the graph Γ, we get E y (cid:16) | Π( X ǫ ( t )) − Π( X ǫ ( s )) | ; t < τ ǫ,η ′ ,δ,δ/ (cid:17) = E y (cid:16) | H ( X ǫ ( t )) − H ( X ǫ ( s )) | ; t < τ ǫ,η ′ ,δ,δ/ (cid:17) ≤ c k ∆ H k ∞ ( t − s ) + sup y ′ ∈{ H ≤ z η ′ } |∇ H ( y ′ ) | ( t − s ) ≤ c T,η ′ ( t − s ) . (5.30)In particular, for every y ∈ D ( ± δ ) | L ǫ,δ ( t, y ) |≤ E y (cid:16) k u ∧ k C (Π( G ( ± δ/ c ) | Π( X ǫ ( t − ǫ α ))) − Π( X ǫ ( t )) | ; t − ǫ α < τ ǫ,η ′ ,δ,δ/ , t < τ ǫ,η ′ ,δ,δ/ (cid:17) ≤ k u ∧ k C (Π( G ( ± δ/ c ) c / T,η ′ ǫ α/ . (5.31)Next, concerning L ǫ,δ ( t, y ), we have P y (cid:16) t − ǫ α < τ ǫ,η ′ ,δ,δ/ , t ≥ τ ǫ,η ′ ,δ,δ/ (cid:17) ≤ P y (cid:16) | H ( X ǫ ( τ ǫ,η ′ ,δ,δ/ )) − H ( X ǫ ( t − ǫ α )) | ≥ δ/ , t − ǫ α < τ ǫ,η ′ ,δ,δ/ , t ≥ τ ǫ,η ′ ,δ,δ/ (cid:17) ≤ P y (cid:16) | H ( X ǫ ( t ∧ τ ǫ,η ′ ,δ,δ/ )) − H ( X ǫ (( t − ǫ α ) ∧ τ ǫ,η ′ ,δ,δ/ )) | ≥ δ/ (cid:17) ≤ δ E y (cid:16) | H ( X ǫ ( t ∧ τ ǫ,η ′ ,δ,δ/ )) − H ( X ǫ (( t − ǫ α ) ∧ τ ǫ,η ′ ,δ,δ/ )) | (cid:17) , (5.32)and, thanks to (5.30), this yields sup y ∈ D ( ± δ ) | L ǫ,δ ( t, y ) | ≤ δ c T,η ′ ǫ α . (5.33)Finally, we consider L ǫ,δ ( t, y ). As a consequence of the Markov property, E y (cid:16) u ( X ǫ ( t )) − ( u ∧ ) ∨ ( X ǫ ( t − ǫ α )) ; t − ǫ α < τ ǫ,η ′ ,δ,δ/ (cid:17) = E y (cid:16) ψ ǫ ( ǫ α , X ǫ ( t − ǫ α )) ; t − ǫ α < τ ǫ,η ′ ,δ,δ/ (cid:17) , ψ ǫ ( s, x ) = E x u ( X ǫ ( s )) − ( u ∧ ) ∨ ( x ) . Since the family { Π( X ǫ ) } ǫ> is weakly convergent in C ([0 , + ∞ ); Γ) and H ( x ) ↑ ∞ , as | x | ↑ ∞ ,we have that for any η > M η > (cid:12)(cid:12)(cid:12) E y (cid:16) ψ ǫ ( ǫ α , X ǫ ( t − ǫ α )) ; t − ǫ α < τ ǫ,η ′ ,δ,δ/ , | X ǫ ( t − ǫ α ) | > M η (cid:17)(cid:12)(cid:12)(cid:12) ≤ k u k ∞ P y ( | X ǫ ( t − ǫ α ) | > M η ) ≤ η. Therefore, | L ǫ,δ ( t, y ) | ≤ η + E y (cid:16) | ψ ǫ ( ǫ α , X ǫ ( t − ǫ α )) | ; t − ǫ α < τ ǫ,η ′ ,δ,δ/ , | X ǫ ( t − ǫ α ) | ≤ M η (cid:17) . As above, we write E y (cid:16) | ψ ǫ ( ǫ α , X ǫ ( t − ǫ α )) | ; t − ǫ α < τ ǫ,η ′ ,δ,δ/ , | X ǫ ( t − ǫ α ) | ≤ M η (cid:17) = E y (cid:16) | ψ ǫ ( ǫ α , X ǫ ( t − ǫ α )) | ; t − ǫ α < τ ǫ,η ′ ,δ,δ/ , t ≥ τ ǫ,η ′ ,δ,δ/ | X ǫ ( t − ǫ α ) | ≤ M η (cid:17) + E y (cid:16) | ψ ǫ ( ǫ α , X ǫ ( t − ǫ α )) | ; t − ǫ α < τ ǫ,η ′ ,δ,δ/ , t < τ ǫ,η ′ ,δ,δ/ | X ǫ ( t − ǫ α ) | ≤ M η (cid:17) =: I ǫ, ( t ) + I ǫ, ( t ) . Due to (5.30) and (5.32), we have I ǫ, ( t ) ≤ k u k ∞ c T ǫ α δ . Now, in [3, Lemma 4.3], it has been proved that, as a consequence of the averaging principle,under the crucial assumption (5.3) given in Hypothesis 3, if α ∈ (4 / , / δ > ǫ → sup x ∈ K | ψ ǫ ( ǫ α , x ) | = lim ǫ → sup x ∈ K (cid:12)(cid:12) E x u ( X ǫ ( ǫ α )) − ( u ∧ ) ∨ ( x ) (cid:12)(cid:12) = 0 , (5.34)for any compact subset K in G ( ± δ/ c and any function u whose support is contained in G ( ± δ/ c . Since X ǫ ( t ) ∈ G ( ± δ/ c , if t < τ ǫ,η ′ ,δ,δ/ , (5.34) implies thatlim ǫ → sup y ∈ D ( ± δ ) E y (cid:16) | ψ ǫ ( ǫ α , X ǫ ( t − ǫ α )) | ; t − ǫ α < τ ǫ,η ′ ,δ,δ/ , t < τ ǫ,η ′ ,δ,δ/ , | X ǫ ( t − ǫ α ) | ≤ M η (cid:17) = 0 , and, because of the arbitrariness of η >
0, we conclude thatlim ǫ → sup t ∈ [ τ,T ] sup y ∈ D ( ± δ ) | L ǫ,δ ( t, y ) | = 0 . This, together with (5.31) and (5.33), implies that for every δ ≤ δ fixedlim ǫ → sup t ∈ [ τ,T ] ∞ X n =1 | J ǫ,η ′ ,δ ,n ( t ) | = 0 . δ ∈ (0 , δ ] such thatsup t ∈ [ τ,T ] X n ∈ N J ǫ,η ′ , ¯ δ ,n ( t ) < η , and then we pick ǫ η > t ∈ [ τ,T ] ∞ X n =1 | J ǫ,η ′ , ¯ δ ,n ( t ) | < η , ǫ ≤ ǫ η , because of (5.27) and (5.28), we can conclude thatsup t ∈ [ τ,T ] (cid:12)(cid:12) E x (cid:0) u ( X ǫ ( t )) − ( u ∧ ) ∨ ( X ǫ ( t )) ; t − ǫ α < ρ ǫ,η ′ (cid:1)(cid:12)(cid:12) < η , ǫ < ǫ η , and (5.26) follows. The first immediate consequence of Theorem 5.3 is that the semigroup S ǫ ( t ) converges to thesemigroup ¯ S ( t ) in H γ , as ǫ ↓ Corollary 6.1.
Under Hypotheses 1, 2 and 3, for every < τ < T and u ∈ H γ we have lim ǫ → sup t ∈ [ τ,T ] | S ǫ ( t ) u − ¯ S ( t ) ∨ u | H γ = lim ǫ → sup t ∈ [ τ,T ] | ( S ǫ ( t ) u ) ∧ − ¯ S ( t ) u ∧ | ¯ H γ = 0 . (6.1) Proof.
First of all, we notice that in view of (3.2), the first limit in (6.1) implies the secondone. So, we will only prove the first limit. We have | S ǫ ( t ) u − ¯ S ( t ) ∨ u | H γ = Z R | S ǫ ( t ) u ( x ) − ¯ S ( t ) ∨ u ( x ) | γ ∨ ( x ) dx. If u ∈ C b ( R ), we have sup t ≥ | S ǫ ( t ) u ( x ) − ¯ S ( t ) ∨ u ( x ) | ≤ | u | ∞ . Hence, since γ ∨ ∈ L ( R ), in view of (5.12) and the dominated convergence theorem, we havethat the first limit in (6.1) is true for every u ∈ C b ( R ). Moreover, in view of Hypothesis 2,we have that | S ǫ ( t ) u | H γ ≤ c T | u | H γ , t ∈ [0 , T ] , ǫ > , (6.2)so that ¯ S ( t ) ∨ u ∈ H γ and | ¯ S ( t ) ∨ u | H γ ≤ c T | u | H γ , t ∈ [0 , T ] . (6.3)Since C c ( R ) is dense in L ( R ) and we assume the weight γ ∨ to be continuous and strictlypositive, we have that C c ( R ) is dense in H γ . Actually, if u ∈ H γ , then u √ γ ∨ ∈ L ( R ).Then, if { u n } n ∈ N is a sequence in C c ( R ), converging to u √ γ ∨ ∈ H γ in L ( R ), we have that27he sequence { u n / √ γ ∨ } n ∈ N converges to u in H γ . Moreover, as u n has compact support and1 / √ γ ∨ is continuous and positive, it follows that { u n / √ γ ∨ } n ∈ N ⊂ C c ( R ).Thus, for any u ∈ H γ , we can fix a sequence { u n } n ∈ N ⊂ C c ( R ) converging to u in H γ .Thanks to (6.3), we get that the sequence { ¯ S ( t ) ∨ u n } n ∈ N is Cauchy in H γ , so that we concludethat ∃ lim n →∞ ¯ S ( t ) ∨ u n =: ¯ S ( t ) ∨ u ∈ H γ , the limit does not depend on the sequence { u n } n ∈ N and and (6.3) holds for every u ∈ H γ .Finally, since we have | S ǫ ( t ) u − ¯ S ( t ) ∨ u | H γ ≤ | S ǫ ( t )( u − u n ) | H γ + | ¯ S ( t ) ∨ ( u − u n ) | H γ + | S ǫ ( t ) u n − ¯ S ( t ) ∨ u n | H γ , according to (6.2) and (6.3), for every η > η η ∈ N such thatsup t ∈ [ τ,T ] | S ǫ ( t ) u − ¯ S ( t ) ∨ u | H γ ≤ η + sup t ∈ [ τ,T ] | S ǫ ( t ) u n η − ¯ S ( t ) ∨ u n η | H γ . This allows to conclude, as u u η ∈ C c ( R ). Remark 6.2.
From the proof of the corollary above, it is clear that from the pointwiseconvergence of S ǫ ( t ) u to ¯ S ( t ) ∨ u , as stated in Theorem 5.3, we cannot conclude that limit (6.1)is also true in L ( R ), as the Lebesgue measure in R is not finite. It is only after introducinga weight that we can prove the convergence in H γ . Corollary 6.3.
Under Hypotheses 1, 2 and 3, we have that the semigroup ¯ S ( t ) is well definedin ¯ H γ and for any T > there exists c T > such that k ¯ S ( t ) k L ( ¯ H γ ) ≤ c T , t ∈ [0 , T ] . (6.4) Proof.
In (6.3) we have seen that ¯ S ( t ) ∨ is well defined in H γ and for any T > c T > u ∈ H γ | ¯ S ( t ) ∨ u | H γ ≤ c T | u | H γ , t ∈ [0 , T ] . Therefore, thanks to (3.3) and (3.2), if f ∈ ¯ H | ¯ S ( t ) f | ¯ H γ = | ( ¯ S ( t ) f ) ∨ | H γ = | ¯ S ( t ) ∨ ( f ∨ ) | H γ ≤ c T | f ∨ | H γ = c T | f | ¯ H γ , t ∈ [0 , T ] , and this allows to conclude. R to the SPDE on the graph We are interested here in the equation ∂ t u ǫ ( t, x ) = L ǫ u ǫ ( t, x ) + b ( u ǫ ( t, x )) + g ( u ǫ ( t, x )) ∂ t W ( t, x ) , t > ,u ǫ (0 , x ) = ϕ ( x ) , x ∈ R . (7.1)28 ǫ is the second order differential operator defined by L ǫ u ( x ) = 12 ∆ u ( x ) + 1 ǫ h ¯ ∇ H ( x ) , ∇ u ( x ) i , x ∈ R , associated with equation (2.11) and with Markov transition semigroup S ǫ ( t ). The Hamiltonian H satisfies Hypotheses 1 and 2 and and the nonlinearities b, g : R → R are assumed to beLipschitz continuous.Concerning the random forcing W , we assume that it is a spatially homogeneous Wienerprocess , with finite spectral measure µ (see e.g. [6] and [2] for all details). This means thatthere exists a Gaussian random field on [0 , + ∞ ) × R , that we also denote by W , defined onsome stochastic basis ( Ω , F , { F t } t ≥ , P ), such that1. the mapping ( t, x )
7→ W ( t, x ) is continuous with respect to t and measurable with respectto both variables, P -almost surely;2. for each x ∈ R , the process W ( t, x ), t ≥
0, is a one-dimensional Wiener process;3. for every t, s ≥ x, y ∈ R E W ( t, x ) W ( s, y ) = ( t ∧ s ) Λ( x − y ) , (7.2)where Λ is the Fourier transform of the spectral measure µ , that isΛ( x ) = Z R e i h x,λ i µ ( dλ ) , x ∈ R . Notice that, with this definition, W ( t, · ) ∈ L ( Ω ; H γ ). Actually, we have E |W ( t, · ) | H γ = E Z R |W ( t, x ) | γ ∨ ( x ) dx = Z R E |W ( t, x ) | γ ∨ ( x ) dx = t Λ(0) Z R γ ∨ ( x ) dx. In what follows, we denote by L s ) ( R , dµ ) the subspace of the Hilbert space L ( R , dµ ; C )consisting of all functions ϕ such that ϕ ( s ) = ϕ , where ϕ ( s ) ( x ) = ϕ ( − x ) , x ∈ R . Moreover, we denote by RK the reproducing kernel Hilbert space of the Wiener process W (see [2] for the definition).As shown in [6, Proposition 1.2], an orthonormal basis for the reproducing kernel RK is given by { d u j µ } j ∈ N , where { u j } j ∈ N is a complete orthonormal basis of the Hilbert space L s ) ( R , dµ ) and d u j µ ( x ) = Z R e i h x,λ i u j ( λ ) µ ( dλ ) , x ∈ R . This means, in particular, that W ( t, x ) can be represented as W ( t, x ) = ∞ X j =1 d u j µ ( x ) β j ( t ) , t ≥ , (7.3)29or some sequence of independent Brownian motions { β j } j ∈ N , all defined on the same stochasticbasis. Moreover, in [6] it is also shown that ∞ X j =1 | d u j µ | H γ < ∞ . (7.4)For every u ∈ H γ and v in the reproducing kernel of W , we shall denote B ( u )( x ) = b ( u ( x )) , G ( u ) v ( x ) = [ g ( u ) v ]( x ) , x ∈ R . Since we are assuming b to be Lipschitz continuous, we have that B : H γ → H γ is Lipschitzcontinuous. Moreover, as proved in [6, Lemma 4.1], G maps H γ into L ( RK, H γ ), where L ( RK, H γ ) is the space of Hilbert-Schmidt operators defined on RK with values in H γ .Furthermore, for every u , u ∈ H γ | G ( u ) − G ( u ) | L ( RK,H γ ) ≤ c | u − u | H γ . (7.5)By using a stochastic factorization argument and (4.1), this implies that E sup t ∈ [0 ,T ] (cid:12)(cid:12)(cid:12)(cid:12)Z t S ǫ ( t − s ) [ G ( u ( s )) − G ( u ( s ))] d W ( s ) (cid:12)(cid:12)(cid:12)(cid:12) pH γ ≤ c p ( T ) E sup t ∈ [0 ,T ] | u ( t ) − u ( t ) | pH γ . (7.6) Definition 7.1.
A predictable process u ǫ ( t ) , taking values in H γ is a mild solution to equation (7.1) if it satisfies the following integral equation u ǫ ( t ) = S ǫ ( t ) ϕ + Z t S ǫ ( t − s ) B ( u ǫ ( s )) ds + Z t S ǫ ( t − s ) G ( u ǫ ( s )) d W ( s ) . (7.7)In [6, Theorem 2.1] it is proved that, under our assumptions on the Hamiltonian H , thecoefficients b and g and the noise W , for every ϕ ∈ H γ and every p ≥ T > u ǫ for equation (7.1) in L p ( Ω ; C ([0 , T ]; H γ )). Moreover, we have E sup t ∈ [0 ,T ] | u ǫ ( t ) | pH γ = c p ( T ) , ǫ > . (7.8)Our purpose is studying the limiting behavior of u ǫ in the space L p ( Ω ; C ([0 , T ]; H γ )) or,equivalently, the limit of u ∧ ǫ in the space L p ( Ω ; C ([0 , T ]; ¯ H γ )), as ǫ ↓ u of the following SPDE on the graph Γ ∂ t ¯ u ( t, z, k ) = ¯ L ¯ u ( t, z, k ) + b (¯ u ( t, z, k )) + g (¯ u ( t, z, k )) ∂ t ¯ W ( t, z, k ) , t > , ¯ u (0 , z, k ) = ϕ ∧ ( z, k ) , ( z, k ) ∈ Γ . (7.9)Here ¯ L is the differential operator on the graph Γ, introduced in Subsection 2.3, generator ofthe limiting Markov process ¯ Y ( t ). Concerning the noisy forcing ¯ W , it is defined by¯ W ( t, z, k ) = ∞ X j =1 ( d u j µ ) ∧ ( z, k ) β j ( t ) , t ≥ z, k ) ∈ Γ , { β j } j ∈ N is the sequence of independent Brownian motions introduced in (7.3). We have E ¯ W ( t, z , k ) ¯ W ( s, z , k ) = E I C k ( z ) W ( t, x ) dµ z ,k I C k ( z ) W ( t, y ) dµ z ,k = I C k ( z ) I C k ( z ) E W ( t, x ) W ( t, y ) dµ z ,k dµ z ,k . Thanks to (7.2), this gives E ¯ W ( t, z , k ) ¯ W ( s, z , k )= ( t ∧ s ) I C k ( z ) I C k ( z ) Z R e i h λ,x − y i µ ( dλ ) dµ z ,k dµ z ,k = ( t ∧ s ) Z R "I C k ( z ) e i h λ,x i dµ z ,k I C k ( z ) e − i h λ,y i dµ z ,k µ ( dλ )= ( t ∧ s ) Z R (cid:16) e i h λ, ·i (cid:17) ∧ ( z , k ) (cid:16) e − i h λ, ·i (cid:17) ∧ ( z , k ) µ ( dλ ) . (7.10)It is immediate to check that (¯ u ) ∧ = u ∧ , for every u : R → C . Hence, thanks to (7.10) and(3.2), this allows to conclude E | ¯ W ( t ) | H γ = n X k =1 Z I k E | ¯ W ( t, z, k ) | T k ( z ) γ ( z, k ) dz = t n X k =1 Z I k Z R (cid:16) e i h λ, ·i (cid:17) ∧ (cid:16) e − i h λ, ·i (cid:17) ∧ ( z, k ) µ ( dλ ) T k ( z ) γ ( z, k ) dz = t Z R " n X k =1 Z I k (cid:12)(cid:12) (cid:16) e i h λ, ·i (cid:17) ∧ (cid:12)(cid:12) ( z, k ) T k ( z ) γ ( z, k ) dz µ ( dλ ) = t Z R (cid:12)(cid:12) (cid:16) e i h λ, ·i (cid:17) ∧ (cid:12)(cid:12) H γ µ ( dλ ) ≤ t Z R (cid:12)(cid:12) e i h λ, ·i (cid:12)(cid:12) H γ µ ( dλ ) = t µ ( R ) Z R γ ∨ ( x ) dx < ∞ . This means that ¯ W ( t ) ∈ L ( Ω ; ¯ H γ ), for every t ≥ ∞ X j =1 | ( d u j µ ) ∧ | H γ ≤ ∞ X j =1 | d u j µ | H γ < ∞ . Then, in view of (6.4), we have that for every v ∈ L p ( Ω ; C ([0 , T ]; ¯ H γ )) the stochastic convo-lution t ∈ [0 , + ∞ ) Z t ¯ S ( t − s ) G ( v ( s )) d ¯ W ( s ) = ∞ X k =1 Z t ¯ S ( t − s ) (cid:2) G ( v ( s ))( d u j µ ) ∧ (cid:3) dβ j ( s ) ,
31s well defined in L ( Ω ; ¯ H γ ) and if v , v ∈ L p ( Ω ; C ([0 , T ]; ¯ H γ )), we have E sup t ∈ [0 ,T ] (cid:12)(cid:12)(cid:12)(cid:12)Z t ¯ S ( t − s ) [ G ( v ( s )) − G ( v ( s ))] d ¯ W ( s ) (cid:12)(cid:12)(cid:12)(cid:12) H γ ≤ c ( T ) E sup t ∈ [0 ,T ] | v ( t ) − v ( t ) | H γ . (7.11) Definition 7.2.
A predictable process ¯ u ( t ) , taking values in ¯ H γ , is a mild solution to equation (7.9) if it satisfies the following integral equation ¯ u ( t ) = ¯ S ( t ) ϕ ∧ + Z t ¯ S ( t − s ) B (¯ u ( s )) ds + Z t ¯ S ( t − s ) G (¯ u ( s )) d ¯ W ( s ) . (7.12)Since the mapping B can be extended to ¯ H γ , as a Lipschitz continuous mapping, dueto (7.11) we can conclude that there exists a unique mild solution ¯ u to equation (7.9) in L p ( Ω , C ([0 , T ]; ¯ H γ )), for any T > p ≥
1. Moreover E sup t ∈ [0 ,T ] | ¯ u ( t ) | p ¯ H γ < + ∞ . (7.13) Theorem 7.3.
Assume Hypotheses 1, 2 and 3. Then, for any ϕ ∈ H γ , p ≥ and < τ < T we have lim ǫ → E sup t ∈ [ τ,T ] | u ǫ ( t ) − ¯ u ( t ) ∨ | pH γ = lim ǫ → E sup t ∈ [ τ,T ] | u ǫ ( t ) ∧ − ¯ u ( t ) | p ¯ H γ = 0 . (7.14) Proof. As u ǫ and ¯ u are mild solutions to equation (7.1) and (7.9), respectively, we have u ǫ ( t ) − ¯ u ( t ) ∨ = (cid:2) S ǫ ( t ) ϕ − ¯ S ( t ) ∨ ϕ (cid:3) + (cid:20)Z t S ǫ ( t − s ) B ( u ǫ ( s )) ds − Z t ¯ S ( t − s ) ∨ B (¯ u ( s ) ∨ ) ds (cid:21) + (cid:2) Θ ǫ ( t ) − ¯Θ ∨ ( t ) (cid:3) =: X i =1 I ǫ,i ( t ) , where, for the sake of brevity, we have definedΘ ǫ ( t ) := Z t S ǫ ( t − s ) G ( u ǫ ( s )) d W ( s ) , ¯Θ( t ) := Z t ¯ S ( t − s ) G (¯ u ( s )) d ¯ W ( s ) . In order to conclude the proof of (7.14), we need a couple of Lemmas, whose proof ispostponed to the end of this section.
Lemma 7.4.
For every p ≥ , T > and < τ ≤ t ≤ T E sup s ∈ [0 ,t ] | I ǫ, ( s ) | pH γ ≤ c p ( T ) Z tτ E sup r ∈ [ τ,s ] | u ǫ ( r ) − ¯ u ( r ) ∨ | pH γ ds + R T,p ( τ, ǫ ) , (7.15) for some constant R T,p ( τ, ǫ ) such that lim ǫ,τ → R T,p ( τ, ǫ ) = 0 . (7.16)32 emma 7.5. For every p ≥ , T > and < τ ≤ t ≤ T E sup s ∈ [0 ,t ] | I ǫ, ( s ) | pH γ ≤ c p ( T ) Z tτ E sup r ∈ [ τ,s ] | u ǫ ( r ) − ¯ u ( r ) ∨ | pH γ ds + S T,p ( τ, ǫ ) , (7.17) for some constant S T,p ( τ, ǫ ) such that lim ǫ,τ → S T,p ( τ, ǫ ) = 0 . (7.18)Now, because of Lemmas 7.4 and 7.5, we have E sup s ∈ [ τ,t ] | u ǫ ( s ) − ¯ u ∨ ( s ) | pH γ ≤ c sup s ∈ [ τ,T ] | S ǫ ( s ) ϕ − ¯ S ( s ) ∨ ϕ | pH γ + R T,p ( τ, ǫ ) + S T,p ( τ, ǫ ) ! + c p ( T ) Z tτ E sup r ∈ [ τ,s ] | u ǫ ( r ) − ¯ u ∨ ( r ) | pH γ ds. Then, thanks the Gronwall lemma E sup s ∈ [ τ,T ] | u ǫ ( s ) − ¯ u ∨ ( s ) | pH γ ≤ c p e c p ( T )( T − τ ) " sup s ∈ [ τ,T ] | S ǫ ( s ) ϕ − ¯ S ( s ) ∨ ϕ | pH γ + R T,p ( τ, ǫ ) + S T,p ( τ, ǫ ) . and we can conclude, thanks to (6.1), (7.16) and (7.18). Proof of Lemma 7.4.
We have I ǫ, ( t ) = Z t S ǫ ( t − s ) (cid:2) B ( u ǫ ( s )) − B (¯ u ( s ) ∨ ) (cid:3) ds + Z t (cid:2) S ǫ ( t − s ) − ¯ S ( t − s ) ∨ (cid:3) B (¯ u ( s ) ∨ ) ds =: J ǫ, ( t ) + J ǫ, ( t ) . (8.1)If we fix 0 < τ ≤ t ≤ T , , we have | J ǫ, ( t ) | pH γ ≤ c p ( T ) (cid:18)Z τ (cid:16) | u ǫ ( s ) | pH γ + | ¯ u ( s ) ∨ | pH γ + 1 (cid:17) ds + Z tτ | u ǫ ( s ) − ¯ u ( s ) ∨ | pH γ ds (cid:19) ≤ c p ( T ) sup s ∈ [0 ,T ] (cid:16) | u ǫ ( s ) | pH γ + | ¯ u ( s ) ∨ | pH γ + 1 (cid:17) τ + c p ( T ) Z tτ sup r ∈ [ τ,s ] | u ǫ ( r ) − ¯ u ( r ) ∨ | pH γ ds. E sup s ∈ [0 ,t ] | J ǫ, ( s ) | pH γ ≤ c p ( T ) τ + c p ( T ) Z tτ E sup r ∈ [ τ,s ] | u ǫ ( r ) − ¯ u ( r ) ∨ | pH γ ds. (8.2)Concerning J ǫ, ( t ), we have | J ǫ, ( t ) | pH γ ≤ c p ( T ) Z t − τ (cid:12)(cid:12)(cid:2) S ǫ ( t − s ) − ¯ S ( t − s ) ∨ (cid:3) B (¯ u ( s ) ∨ ) (cid:12)(cid:12) pH γ ds + c p ( T ) Z tt − τ (cid:16) | ¯ u ( s ) ∨ | pH γ (cid:17) ds ≤ c p ( T ) Z T sup r ∈ [ τ,T ] (cid:12)(cid:12)(cid:2) S ǫ ( r ) − ¯ S ( r ) ∨ (cid:3) B (¯ u ( s ) ∨ ) (cid:12)(cid:12) pH γ ds + c p ( T ) s ∈ [0 ,T ] | ¯ u ( s ) ∨ | pH γ ! τ. Then, according to (7.13), we conclude E sup t ∈ [0 ,T ] | J ǫ, ( t ) | pH γ ≤ c p ( T ) τ + E Z T sup r ∈ [ τ,T ] (cid:12)(cid:12)(cid:2) S ǫ ( r ) − ¯ S ( r ) ∨ (cid:3) B (¯ u ( s ) ∨ ) (cid:12)(cid:12) pH γ ds ! . This, together with (8.2), implies (7.15), with R T,p ( τ, ǫ ) := c p ( T ) τ + E Z T sup r ∈ [ τ,T ] (cid:12)(cid:12)(cid:2) S ǫ ( r ) − ¯ S ( r ) ∨ (cid:3) B (¯ u ( s ) ∨ ) (cid:12)(cid:12) pH γ ds. Moreover, (7.16) follows as a consequence of Theorem 5.3 and the dominated convergencetheorem, due to the equi-boundedness of the norm of S ǫ ( t ) and ¯ S ( t ), and to (7.13). Proof of Lemma 7.5.
First of all, we notice that, due to (3.5),¯Θ ∨ ( t ) := Z t ¯ S ( t − s ) ∨ G (¯ u ( s ) ∨ ) d W ( s ) . Then I ǫ, ( t ) = Z t S ǫ ( t − s ) (cid:2) G ( u ǫ ( s )) − G (¯ u ( s ) ∨ ) (cid:3) d W ( s )+ Z t (cid:2) S ǫ ( t − s ) − ¯ S ( t − s ) ∨ (cid:3) G (¯ u ( s ) ∨ ) d W ( s ) =: J ǫ, ( t ) + J ǫ, ( t ) . By using a factorization argument, for every t ∈ [0 , T ] and α ∈ (0 , /
2) we have J ǫ, ( t ) = sin παπ Z t ( t − s ) α − S ǫ ( t − s ) Y α ( s ) ds, Y α ( s ) = Z s ( s − σ ) − α S ǫ ( s − σ ) (cid:2) G ( u ǫ ( σ )) − G (¯ u ( σ ) ∨ ) (cid:3) d W ( σ ) . Therefore, thanks to (4.1), for every p ≥ /α E sup s ∈ [0 ,t ] | J ǫ, ( s ) | pH γ ≤ c p,α ( T ) Z t E | Y α ( s ) | pH γ ds. (8.3)In view of (4.1) and (7.5), we have E | Y α ( s ) | pH γ = c p E Z s ( s − σ ) − α ∞ X j =1 (cid:12)(cid:12) S ǫ ( s − σ ) (cid:2) G ( u ǫ ( σ )) − G (¯ u ( σ ) ∨ ) (cid:3) d u j µ (cid:12)(cid:12) H γ dσ p ≤ c p ( T ) E Z s ( s − σ ) − α ∞ X j =1 (cid:12)(cid:12)(cid:2) G ( u ǫ ( σ )) − G (¯ u ( σ ) ∨ ) (cid:3) d u j µ (cid:12)(cid:12) H γ dσ p = c p ( T ) E (cid:18)Z s ( s − σ ) − α k G ( u ǫ ( σ )) − G (¯ u ( σ ) ∨ ) k L ( RK,H γ ) dσ (cid:19) p ≤ c p ( T ) E (cid:18)Z s ( s − σ ) − α | u ǫ ( σ ) − ¯ u ( σ ) ∨ | H γ dσ (cid:19) p . Hence, thanks to the Young inequality, due to (7.8) and (7.13), for any 0 < τ < t < T Z t E | Y α ( s ) | pH γ ds ≤ c p ( T ) (cid:18)Z t s − α ds (cid:19) p Z t E | u ǫ ( s ) − ¯ u ( s ) ∨ | pH γ ds ≤ c p ( T ) τ E sup s ∈ [0 ,T ] | u ǫ ( s ) | pH γ + E sup s ∈ [0 ,T ] | ¯ u ( s ) | pH γ ! + c p ( T ) Z tτ E sup r ∈ [ τ,s ] | u ǫ ( r ) − ¯ u ( r ) ∨ | pH γ ds ≤ c p ( T ) τ + Z tτ E sup r ∈ [ τ,s ] | u ǫ ( r ) − ¯ u ( r ) ∨ | pH γ ds ! . According to (8.3), this yields E sup s ∈ [0 ,t ] | J ǫ, ( s ) | pH γ ≤ c p ( T ) τ + Z tτ E sup r ∈ [ τ,s ] | u ǫ ( r ) − ¯ u ( r ) ∨ | pH γ ds ! . (8.4)Next, by using again a factorization argument, for every t ∈ [0 , T ] and α ∈ (0 , /
2) wehave J ǫ, ( t ) = sin παπ Z t ( t − s ) α − S ǫ ( t − s ) Y α, ( s ) ds + sin παπ Z t ( t − s ) α − (cid:2) S ǫ ( t − s ) − ¯ S ( t − s ) ∨ (cid:3) Y α, ( s ) ds, Y α, ( s ) = Z s ( s − σ ) − α (cid:2) S ǫ ( s − σ ) − ¯ S ( s − σ ) ∨ (cid:3) G (¯ u ( σ ) ∨ ) d W ( σ ) , and Y α, ( s ) = Z s ( s − σ ) − α ¯ S ( s − σ ) ∨ G (¯ u ( σ ) ∨ ) d W ( σ ) . Thus, by proceeding as above for J ǫ, ( t ), thanks to (4.1) we get E sup s ∈ [0 ,t ] | J ǫ, ( s ) | pH γ ≤ c p,α ( T ) Z t E | Y α, ( s ) | pH γ ds + c p,α ( T ) E sup s ∈ [0 ,t ] Z s (cid:12)(cid:12)(cid:2) S ǫ ( s − r ) − ¯ S ( s − r ) ∨ (cid:3) Y α, ( r ) (cid:12)(cid:12) pH γ dr =: H ǫ, ( t ) + H ǫ, ( t ) . According to (4.1) and (6.4), we have E | Y α, ( s ) | pH γ ≤ c p E ∞ X j =1 Z s ( s − σ ) − α (cid:12)(cid:12)(cid:2) S ǫ ( s − σ ) − ¯ S ( s − σ ) ∨ (cid:3) G (¯ u ( σ ) ∨ ) d u j µ (cid:12)(cid:12) H γ dσ p ≤ c p E ∞ X j =1 Z s ( s − σ ) − α (cid:12)(cid:12) G (¯ u ( σ ) ∨ ) d u j µ (cid:12)(cid:12) H γ dσ p . Since ∞ X j =1 Z s ( s − σ ) − α (cid:12)(cid:12) G (¯ u ( σ ) ∨ ) d u j µ (cid:12)(cid:12) H γ dσ p ≤ c p ( T ) σ ∈ [0 ,s ] | ¯ u ( σ ) ∨ | pH γ ! , due to (7.13) and the dominated convergence theorem, we havelim n →∞ E ∞ X j = n +1 Z s ( s − σ ) − α (cid:12)(cid:12) G (¯ u ( σ ) ∨ ) d u j µ (cid:12)(cid:12) H γ dσ p = 0 . Therefore, for every η > n η ∈ N such that E | Y α, ( s ) | pH γ ≤ η + c p ( T ) E ∞ X j = n η +1 Z s ( s − σ ) − α (cid:12)(cid:12)(cid:2) S ǫ ( s − σ ) − ¯ S ( s − σ ) ∨ (cid:3) G (¯ u ( σ ) ∨ ) d u j µ (cid:12)(cid:12) H γ dσ p . Once fixed n η , due to (6.1) and the dominated convergence theorem, we havelim ǫ → E ∞ X j = n η +1 Z s ( s − σ ) − α (cid:12)(cid:12)(cid:2) S ǫ ( s − σ ) − ¯ S ( s − σ ) ∨ (cid:3) G (¯ u ( σ ) ∨ ) d u j µ (cid:12)(cid:12) H γ dσ p = 0 , η , allows us to conclude thatlim ǫ → sup t ∈ [0 ,T ] H ǫ, ( t ) = 0 . (8.5)As far H ǫ, ( t ) is concerned, due to (4.1) and (6.4) for every 0 < τ ≤ s ≤ t we have Z s (cid:12)(cid:12)(cid:2) S ǫ ( s − r ) − ¯ S ( s − r ) ∨ (cid:3) Y α, ( r ) (cid:12)(cid:12) pH γ dr = Z s − τ (cid:12)(cid:12)(cid:2) S ǫ ( s − r ) − ¯ S ( s − r ) ∨ (cid:3) Y α, ( r ) (cid:12)(cid:12) pH γ dr + Z ss − τ (cid:12)(cid:12)(cid:2) S ǫ ( s − r ) − ¯ S ( s − r ) ∨ (cid:3) Y α, ( r ) (cid:12)(cid:12) pH γ dr ≤ Z T sup ρ ∈ [ τ,T ] (cid:12)(cid:12)(cid:2) S ǫ ( ρ ) − ¯ S ( ρ ) ∨ (cid:3) Y α, ( r ) (cid:12)(cid:12) pH γ dr + c p ( T ) √ τ (cid:18)Z T | Y α, ( r ) | pH γ dr (cid:19) . Moreover, if 0 ≤ s ≤ τ , we have Z s (cid:12)(cid:12)(cid:2) S ǫ ( s − r ) − ¯ S ( s − r ) ∨ (cid:3) Y α, ( r ) (cid:12)(cid:12) pH γ dr ≤ c p ( T ) √ τ (cid:18)Z T | Y α, ( r ) | pH γ dr (cid:19) . Since, for every q ≥ E Z T | Y α, ( r ) | qH γ dr < ∞ , (8.6)this implies H ǫ, ( t ) ≤ E Z T sup ρ ∈ [ τ,T ] (cid:12)(cid:12)(cid:2) S ǫ ( ρ ) − ¯ S ( ρ ) ∨ (cid:3) Y α, ( r ) (cid:12)(cid:12) pH γ dr + c p ( T ) √ τ . Now, let us fix any η > τ η > c p ( T ) √ τ η < η . Because of (6.1), we havelim ǫ → sup ρ ∈ [ τ η ,T ] (cid:12)(cid:12)(cid:2) S ǫ ( ρ ) − ¯ S ( ρ ) ∨ (cid:3) Y α, ( r ) (cid:12)(cid:12) pH γ = 0 . Moreover, as sup ρ ∈ [ τ η ,T ] (cid:12)(cid:12)(cid:2) S ǫ ( ρ ) − ¯ S ( ρ ) ∨ (cid:3) Y α, ( r ) (cid:12)(cid:12) pH γ ≤ c p ( T ) | Y α, ( r ) | pH γ , from (8.6) and the dominated convergence theorem, we getlim inf ǫ → sup t ∈ [0 ,T ] H ǫ, ( t ) ≤ lim ǫ → E Z T sup ρ ∈ [ τ,T ] (cid:12)(cid:12)(cid:2) S ǫ ( ρ ) − ¯ S ( ρ ) ∨ (cid:3) Y α, ( r ) (cid:12)(cid:12) pH γ dr + η = η. Due to the arbitrariness of η >
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