Stability theory for Gaussian rough differential equations. Part I
aa r X i v : . [ m a t h . P R ] J a n Stability theory for Gaussian rough differential equations. Part I.
Luu Hoang DucMax-Planck-Institut f¨ur Mathematik in den Naturwissenschaften, &Institute of Mathematics, Vietnam Academy of Science and Technology
E-mail: [email protected], [email protected]
Abstract
We propose a quantitative direct method of proving the stability result for Gaussian roughdifferential equations in the sense of Gubinelli [22]. Under the strongly dissipative assumption ofthe drift coefficient function, we prove that the trivial solution of the system under small noiseis exponentially stable.
Keywords: stochastic differential equations (SDE), Young integral, rough path theory, roughdifferential equations, exponential stability.
This paper deal with the asymptotic stability criteria for rough differential equations of the form dy t = [ Ay t + f ( y t )] dt + G ( y t ) dx t , (1.1)or in the integral form y t = y a + Z ta [ Ay u + f ( y u )] du + Z ta G ( y u ) dx u , t ∈ [ a, T ]; (1.2)where the nonlinear part f : R d → R d is globally Lipschitz function for simplicity and G =( G , . . . , G m ) is a collection of vector fields G j : R d → R d such that G ( y ) = g ( y ) , where g = ( g , . . . , g m ) , g j ∈ C if ν ∈ ( , Cy, where C = ( C , . . . , C m ) , C j ∈ R d × d , if ν ∈ ( , ) g ( y ) , where g = ( g , . . . , g m ) , g j ∈ C b ( R d , R d ) , if ν ∈ ( , ) . (1.3)Equation (1.1) can be viewed as a controlled differential equation driven by rough path x ∈ C ν ([ a, T ] , R m ) for ν ∈ ( , x can also be considered asan element of the space C p − var ([ a, T ] , R m ) of finite p - variation norm, with pν ≥
1. For instance,given ¯ ν ∈ ( , x might be a realization of a R m -valued centered Gaussian processsatisfying: there exists for any T > C T such that for all p ≥ ν E k X t − X s k p ≤ C T | t − s | p ¯ ν , ∀ s, t ∈ [0 , T ] . (1.4)By Kolmogorov theorem, for any ν ∈ (0 , ¯ ν ) and any interval [0 , T ] almost all realization of X willbe in C ν ([0 , T ]). Such a stochastic process, in particular, can be a fractional Brownian motion B H [35] with Hurst exponent H ∈ ( , B H = { B Ht } t ∈ R with continuous sample pathsand E k B Ht − B Hs k = | t − s | H , ∀ t, s ∈ R .
1n this paper, we would like to approach system (1.1), where the second integral is well-understoodas rough integral in the sense of Gubinelli [22]. Such system satisfies the existence and uniqueness ofsolution given initial conditions, see e.g. [22] or [15] for a version without drift coefficient function,and [39] for a full version using p - variation norms.Notice that the question for global asymptotic dynamics of system (1.1) is studied in [6], [25],[26], [27], [28], under the general dissipativity condition for the drift coefficient function, in whichthey prove that there exists a unique smooth stationary density for (1.1), with convergence rate iseither exponential or polynomial, depending on Hurst index H .Meanwhile, the topic of asymptotic stability for pathwise solution of (1.1) is studied in [13] forwhich the noise is assumed to be fractional Brownian motion with small intensity. In addition, thelocal stability is studied in [19] and in [21] for which the diffusion coefficient g ( x ) is rather flat,i.e. g (0) = D y g (0) = 0 for the Young differential equations and g (0) = D y g (0) = D yy g (0) = 0for the rough differential equations. In all mentioned references, the technique in use is semigrouptechnique together with the help of fractional calculus.To study the local stability, we impose conditions for matrices A ∈ R d × d such that A is negativedefinite, i.e. there exists a λ A > h y, Ay i ≤ − λ A k y k . (1.5)The strong condition (1.5) is still able to cover interesting cases, for instance all matrices withnegative real part eigenvalues, under a transformation, since there exists a positive definite matrix Q which is the solution of the matrix equation A T Q + Q A = D where D is a symmetric negative definite matrix [3, Chapter 2 & Chapter 5].To study the local stability, we will assume that the nonlinear part f : R d → R d is locally Lipschitzfunction such that f (0) = 0 and k f ( y ) k ≤ k y k h ( k y k ) (1.6)where h : R + → R + is an increasing function which is bounded above by a constant C f . Ourassumption is somehow still global, but it has an advantage of being able to treat the local dynamicsas well. We refer to [19] and [21] for real local versions on a small neighborhood B (0 , ρ ) of the trivialsolution, using the cutoff technique.In this paper, we also assume that g (0) = 0 and g ∈ C b in case ν ∈ ( , ) with bounded derivatives C g (which also include the Lipschit coefficient of the highest derivative). System (1.1) then admitsan equilibrium which is the trivial solution. Our main stability results are then formulated asfollows. Theorem 1.1 (Stability for Young systems)
Assume X · ( ω ) is a Gaussian process satisfying (1.4) , and ¯ ν > ν > is fixed. Assume further that conditions (1.5) , (1.6) are satisfied, where λ A > h (0) . Then there exists an ǫ > such that given C g < ǫ , and for almost sure all realizations x · = X · ( ω ) , the zero solution of (1.1) is locally exponentially stable. If in addition λ A > C f , thenwe can choose ǫ so that the zero solution of (1.1) is globally exponentially stable a.s. Theorem 1.2 (Stability for rough systems)
Assume X · ( ω ) is a Gaussian process satisfying (1.4) , and > ¯ ν > ν > is fixed. Assume further that G ( y ) = Cy and conditions (1.5) , (1.6) are satisfied, where λ A > h (0) . Then the conclusion of Theorem 1.1 on local stability of the zerosolution holds for almost sure all realizations x of X . If in addition λ A > C f , then we can choose ǫ so that the zero solution of (1.1) is globally exponentially stable a.s. G ( y ) = Cy , since it is not necessary to prove the integrability of ||| θ, θ ′ ||| x, α, [ a,b ] in order to get thepathwise stability.Part II [12] is to present the result for the case G ( y ) = g ( y ) ∈ C b , for which a necessary assumptionis the integrability of solution. This assumption is straightforward for Young equations but nottrivial for the rough case, and even difficult to prove under the H¨older norm. Specifically, theconcept of greedy times for H¨older norms and similar result to [5, Theorem 6.3] on the main tailestimate of the number of greedy time under the α -H¨older norm is not easy to prove. Fortunately,we can overcome this issue by studying Gubinelli approach under the modified ( p − σ ) - variationseminorms in order to apply [5, Theorem 6.3] directly.We close the introduction part with a note that our method still works for the case ν ∈ ( , ]with an extension of Gubinelli derivative to the second order, although the computation would berather complicated. Moreover, it could also be applied for proving the general case in which g isunbounded, even though we then need to prove the existence and uniqueness theorem and also theintegrability of the solution. The reader is referred to [32] and [9] for this approach, in which thedifferential equation is understood in the sense of Davie [11]. ν ∈ ( , : Young differential equations We would like to give a brief introduction to Young integrals. Given any compact time interval I ⊂ R , let C ( I, R d ) denote the space of all continuous paths y : I → R d equipped with supnorm k · k ∞ ,I given by k y k ∞ ,I = sup t ∈ I k y t k , where k · k is the Euclidean norm in R d . We write y s,t := y t − y s . For p ≥
1, denote by C p − var ( I, R d ) ⊂ C ( I, R d ) the space of all continuous path y : I → R d which is of finite p -variation ||| y ||| p -var ,I := sup Π( I ) n X i =1 k y t i ,t i +1 k p ! /p < ∞ , (2.1)where the supremum is taken over the whole class of finite partition of I . C p − var ( I, R d ) equippedwith the p − var norm k y k p -var ,I := k y min I k + ||| y ||| p − var , I , is a nonseparable Banach space [17, Theorem 5.25, p. 92]. Also for each 0 < α <
1, we denote by C α ( I, R d ) the space of H¨older continuous functions with exponent α on I equipped with the norm k y k α,I := k y min I k + ||| y ||| α,I = k y ( a ) k + sup s
1, the Young integral3 I y t dx t can be defined as Z I y s dx s := lim | Π |→ X [ u,v ] ∈ Π y u x u,v , where the limit is taken on all the finite partition Π = { min I = t < t < · · · < t n = max I } of I with | Π | := max [ u,v ] ∈ Π | v − u | (see [40, p. 264–265]). This integral satisfies additive property by theconstruction, and the so-called Young-Loeve estimate [17, Theorem 6.8, p. 116] (cid:13)(cid:13)(cid:13) Z ts y u dx u − y s x s,t (cid:13)(cid:13)(cid:13) ≤ K ( p, q ) ||| y ||| q -var , [ s,t ] ||| x ||| p -var , [ s,t ] ≤ K ( p, q ) | t − s | p + q ||| y ||| p , [ s,t ] ||| x ||| q − Hol , [ s,t ] , (2.2)for all [ s, t ] ⊂ I , where K ( p, q ) := (1 − − p − q ) − . (2.3) Theorem 2.1 (Existence, uniqueness and integrability of the solution)
Under assumptions( H ), ( H ), there exists a unique solution of equation (1.1) on any interval [ a, b ] . Moreover ||| y ||| q − var , [ a,b ] is integrable.Proof: Since ν > , (1.1) is a Young equation which satisfies the assumptions of Theorem3.6 and Theorem 4.4 in [7] on the existence and uniqueness of solution for (1.1) and its backwardequation. Moreover to estimate ||| x ||| q − var , [ a,b ] , we apply [7, Lemma 3.3] to conclude that there existsa function F ( ||| x ||| p − var , [ a,b ] ) = 4 p (log 2) max {k A k + C f , ( K + 1) C g } h ( b − a ) p + ||| x ||| pp − var , [ a,b ] i such that ||| y ||| q − var , [ a,b ] ≤ k y a k exp n F ( ||| x ||| p − var , [ a,b ] ) o k y k ∞ , [ a,b ] ≤ ||| y ||| q − var , [ a,b ] + k y a k ≤ k y a k (cid:16) n F ( ||| x ||| p − var , [ a,b ] ) o(cid:17) . (2.4)From [37] (see also [29, Proposition 2.1,p.18]) the random variable Z := e ||| x ||| p − var , [0 , , with 1 < p < x is a realization of Gaussian stochastic process. Thatproves the integrability of ||| y ||| q − var , [ a,b ] and k y k ∞ , [ a,b ] . Notice that the integrability of ||| y ||| q − var , [ a,b ] and k y k ∞ , [ a,b ] can also be proved using [5, Theorem 6.3] with better estimates. ν ∈ ( , ) : controlled differential equations We also introduce the construction of the integral using rough paths for the case y, x ∈ C α ( I ) when α ∈ ( , ν ). To do that, we need to introduce the concept of rough paths. Following [15], a couple x = ( x, X ), with x ∈ C α ( I, R m ) and X ∈ C α (∆ ( I ) , R m ⊗ R m ) := { X : sup s
Assume that α > , V ∈ C b ( R d , R ) and y ∈ C α ( I, R ) is a solution of the rough differential equation y t = y s + Z ts f ( y u ) du + Z ts g ( y u ) dx u , ∀ min I ≤ s ≤ t ≤ max I. (2.9) Then one get the change of variable formula V ( y t ) = V ( y s ) + Z ts h D y V ( y u ) , f ( y u ) i du + Z ts h D y V ( y u ) g ( y u ) i dx u + 12 Z ts D yy V ( y u )[ g ( y u ) , g ( y u )] d [ x ] s,u , (2.10) where [ D y V ( y ) g ( y )] ′ s = h D y V ( y s ) , D y g ( y s ) g ( y s ) i + D yy V ( y s )[ g ( y s ) , g ( y s )] . Proof:
Using the Taylor expansion, it is easy to see that V ( y t ) = V ( y s ) + h D y V ( y s ) , y s,t i + 12 D yy V ( y s )[ y s,t , y s,t ] + O ( | t − s | α ) . On the other hand, it follows from (2.9) and (2.8) that y s,t = f ( y s )( t − s ) + g ( y s ) x s,t + [ g ( y )] ′ s X s,t + O ( | t − s | α )= f ( y s )( t − s ) + g ( y s ) x s,t + D y g ( y s ) g ( y s ) X s,t + O ( | t − s | α ) . As the result, V ( y ) s,t = h D y V ( y s ) , f ( y s ) i ( t − s ) + h D y V ( y s ) , g ( y s ) i x s,t + D y V ( y s ) D y g ( y s ) g ( y s ) X s,t + 12 D yy V ( y s )[ g ( y s ) , g ( y s )] x s,t ⊗ x s,t + O ( | t − s | α )= h D y V ( y s ) , f ( y s ) i ( t − s ) + h D y V ( y s ) , g ( y s ) i x s,t + 12 D yy V ( y s )[ g ( y s ) , g ( y s )][ x ] s,t + (cid:16) D y V ( y s ) D y g ( y s ) g ( y s ) + D yy V ( y s )[ g ( y s ) , g ( y s )] (cid:17) X s,t + O ( | t − s | α )= h D y V ( y s ) , f ( y s ) i ( t − s ) + h D y V ( y s ) , g ( y s ) i x s,t + [ D y V ( y ) g ( y )] ′ s X s,t + 12 D yy V ( y s )[ g ( y s ) , g ( y s )][ x ] s,t + O ( | t − s | α ) , which is the discretization version of (2.10). The conclusion is then a direct consequence of thesewing lemma. 6 .2.2 Greedy times For any ν ∈ ( , ) and on each compact interval I such that | I | = max I − min I = 1, consider arough path x = ( x, X ) ∈ C ν ( I ) with H¨older norm. Then given α ∈ ( , ν ), we construct for anyfixed γ ∈ (0 ,
1) the sequence of greedy times { τ i ( γ, I, α ) } i ∈ N w.r.t. H¨older norms τ = min I, τ i +1 := inf n t > τ i : ||| x ||| α, [ τ i ,t ] = γ o ∧ max I. (2.11)Denote by N γ,I,α ( x ) := sup { i ∈ N : τ i ≤ max I } . From the definition (2.11), it follows that γ < ( τ i +1 − τ i ) ν − α (cid:16) ||| x ||| ν,I + ||| X ||| ν, ∆ ( I ) (cid:17) , which implies that | I | ≥ τ N γ,I,α ( x ) − min I = N γ,I,α ( x ) − X i =0 ( τ i +1 − τ i ) ≥ N γ,I,α ( x ) γ ν − α (cid:16) ||| x ||| ν,I + ||| X ||| ν, ∆ ( I ) (cid:17) − ν − α . This proves that N γ,I,α ( x ) ≤ | I | γ − ν − α (cid:16) ||| x ||| ν,I + ||| X ||| ν, ∆ ( I ) (cid:17) ν − α . (2.12)Also, we construct another sequence of greedy time { ¯ τ i ( γ, I, α ) } i ∈ N given by¯ τ = min I, ¯ τ i +1 := inf n t > ¯ τ i : ( t − ¯ τ i ) − α + ||| x ||| α, [¯ τ i ,t ] = γ o ∧ max I, (2.13)and denote by ¯ N γ,I,α ( x ) := sup { i ∈ N : ¯ τ i ≤ max I } . Then on any interval J such that | J | = (cid:16) γ (cid:17) − α and with the sequence { τ i ( γ , J, α ) } i ∈ N it follows that( τ i +1 − τ i ) − α + ||| x ||| α, [ τ i ,τ i +1 ] ≤ γ γ γ, hence there is a most one greedy time of the sequence ¯ τ i lying in each interval [ τ i ( γ , J, α ) , τ i +1 ( γ , J, α )].That being said, if we divide I into sub-interval J k of length | J k | ≡ | J | = (cid:16) γ (cid:17) − α , then it followsthat ¯ N γ,I,α ( x ) ≤ m X k =1 N γ ,J k ,α ( x ) , m := l | I || J | m . (2.14) Theorem 2.3 (Existence and uniqueness of the solution)
Assume that G ( y ) = Cy , thereexists a unique solution of equation (1.1) and also of the backward equation on any interval [ a, b ] .Proof: To make our presentation self contained, we give a direct proof here for the roughdifferential equation dy = [ Ay t + f ( y t )] dt + Cy t dx t = F ( y t ) dt + Cy t dx t , or in the integral form y t = G ( y, y ′ ) t = y a + Z ta F ( y u ) du + Z ta Cy u dx u , t ∈ [ a, T ] , (2.15)where F (0) is globally Lipschitz continuous with Lipschitz coefficient L f = k A k + C f . Denote by D αx ( y a , Cy a ) the set of paths ( y, y ′ ) controlled by x in [ a, T ] with y a and y ′ a = Cy a fixed. Considerthe mapping defined by M : D αx ( y a , Cy a ) → D αx ( y a , Cy a ) , M ( y, y ′ ) t := ( G ( y, y ′ ) t , Cy t ) . |||M ( y, y ′ ) ||| x, α = ||| Cy ||| α + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R F ( y,y ′ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α using ||| ( y, y ′ ) ||| x, α = ||| y ′ ||| α + ||| R y ||| α . Since ||| Cy ||| α ≤ k C k ||| y ||| α ≤ k C k (cid:16) k y ′ k ∞ ||| x ||| α + ( T − a ) α ||| R y ||| α (cid:17) ≤ k C k ||| x ||| α k y ′ a k + k C k ( T − a ) α ||| x ||| α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α + k C k ( T − a ) α ||| R y ||| α and k R F ( y,y ′ ) s,t k ≤ (cid:13)(cid:13)(cid:13) Z ts F ( y u ) du (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) Z ts Cy u dx u − Cy s x s,t (cid:13)(cid:13)(cid:13) ≤ L f | t − s |k y k ∞ , [ s,t ] + k C kk y ′ k ∞ , [ s,t ] | X s,t | + C α | t − s | α h ||| x ||| α, [ s,t ] k C k ||| R y ||| α, [ s,t ] + k C k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α, [ s,t ] ||| X ||| α, ∆ ([ s,t ]) i , where we can choose T − a < C α can be bounded from above by C α (1). In addition k y k ∞ , [ s,t ] ≤ k y a k + k y ′ a k ( T − a ) α ||| x ||| α + ( T − a ) α ||| R y ||| α , thus it follows that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R F ( y,y ′ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α ≤ ( T − a ) − α L f k y a k + ( T − a ) − α L f ||| x ||| α k y ′ a k + L f ( T − a ) ||| R y ||| α + k C k ||| X ||| α ( | y ′ a | + ( T − a ) α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α ) + C α k C k ( T − a ) α h ||| x ||| α ||| R y ||| α + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α ||| X ||| α i All in all, we can estimate |||M ( y, y ′ ) ||| x, α as follows (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M ( y, y ′ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x, α ≤ k C k h ( k y ′ a k + ( T − a ) α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α ) ||| x ||| α + ( T − a ) α ||| R y ||| α i + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R F ( y,y ′ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α ≤ ( T − a ) − α L f k y a k + h(cid:16) k C k + ( T − a ) − α L f (cid:17) ||| x ||| α + k C k ||| X ||| α i k Cy a k + h ( T − a ) α k C k ||| x ||| α + ( T − a ) α k C k (1 + C α ) ||| X ||| α i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α + h k C k ( T − a ) α + ( T − a ) L f + C α k C k ( T − a ) α ||| x ||| α i ||| R y ||| α ≤ (cid:16) L f + k C k + k C k C α (cid:17) (1 + k C k ) µ k y a k + h L f + k C k + k C k C α i µ (cid:16) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α + ||| R y ||| α (cid:17) ≤ µ (cid:16) k y a k + k Cy a k + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y, y ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x, α (cid:17) where we choose for a fixed number µ ∈ (0 ,
1) with M := max nh L f + k C k (1 + C α ) i (1 + k C k ) , o and T = T ( a ) satisfying( T − a ) − α + ||| x ||| α, [ a,T ] + ||| X ||| α, ∆ ([ a,T ]) = µ M < . Therefore, if we restrict to the set B := n ( y, y ′ ) ∈ D αx ( y a , Cy a ) , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y, y ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x, α ≤ µ − µ k y a k o (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M ( y, y ′ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x, α ≤ µ k y, y ′ k x, α ≤ (cid:16) µ − µ + µ (cid:17) k y a k ≤ µ − µ k y a k , which proves that M : B → B . By Schauder-Tichonorff theorem, there exists a fixed point of M which is a solution of equation (1.1) on the interval [ a, T ]. Next, for any two solutions ( y, y ′ ) , (¯ y, ¯ y ′ )of the same initial conditions ( y a , Cy a ), by similar computations, one get (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( y, y ′ ) − (¯ y, ¯ y ′ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x, α ≤ µ (cid:16) k y a − ¯ y a k + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( y, y ′ ) − (¯ y, ¯ y ′ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x, α (cid:17) ≤ µ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( y, y ′ ) − (¯ y, ¯ y ′ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x, α and together with µ <
1, this proves the uniqueness of solution of (1.1) on [ a, T ]. By constructingthe greedy time sequence (2.13), we can extend and prove the existence of the unique solution onthe whole real line. It is easy to see that solution y t depends linearly on initial y a , hence there existsa solution matrix Φ( t, a, x, X ) of equation (2.15). The similar conclusion holds for the backwardequation.The estimate of the solution under the supremum norm k · k ∞ and the |||· , ·||| x, α semi-norm is provedstraight forward. Theorem 2.4
Assume G ( y ) = Cy . For any interval [ a, b ] , the seminorm ||| y, y ′ ||| x, α, [ a,b ] and thesupremum norm k y k ∞ , [ a,b ] are estimated as follows. (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y, y ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x, α, [ a,b ] ≤ k y a k exp n ¯ N µM , [ a,b ] ,α ( x ) log (cid:16) µ + 11 − µ (cid:17)o ; (2.16) k y k ∞ , [ a,b ] ≤ k y a k exp n ¯ N µM , [ a,b ] ,α ( x ) log (cid:16) µ + 11 − µ (cid:17)o . (2.17) Proof:
To estimate ||| y, y ′ ||| x, α , we use the same greedy time (2.13) to get (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y, y ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x, α, [¯ τ i , ¯ τ i +1 ] ≤ µ − µ k y ¯ τ i k so that k y ¯ τ i +1 k ≤ k y k ∞ , [ τ i ,τ i +1 ] ≤ k y ¯ τ i k + k C kk y ¯ τ i k (¯ τ i +1 − ¯ τ i ) α ||| x ||| α + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y, y ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x, α, [¯ τ i , ¯ τ i +1 ] ≤ (cid:16) k C k (¯ τ i +1 − ¯ τ i ) α ||| x ||| α + µ − µ (cid:17) k y ¯ τ i k ≤ (cid:16) µ + 11 − µ (cid:17) k y ¯ τ i k . (2.18)As a result (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y, y ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x, α, [¯ τ i , ¯ τ i +1 ] ≤ µ − µ k y ¯ τ i k ≤ µ − µ (cid:16) µ + 11 − µ (cid:17) i k y a k and therefore (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y, y ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x, α, [ a,b ] ≤ ¯ N µM , [ a,b ] ,α ( x ) X i =0 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y, y ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x, α, [¯ τ i , ¯ τ i +1 ] ≤ ¯ N µM , [ a,b ] ,α ( x ) X i =0 µ − µ (cid:16) µ + 11 − µ (cid:17) i k y a k≤ k y a k exp n ¯ N µM , [ a,b ] ,α ( x ) log (cid:16) µ + 11 − µ (cid:17)o . The same estimate using (2.18) shows (2.17). 9
Stability results
We first present the definition of pathwise stability (see e.g. [14]).
Definition 3.1 (A) Stability: A solution µ ( · ) of the deterministic differential equation (1.1) iscalled stable, if for any ε > there exists an r = r ( ε ) > such that for any solution y of (1.1) satisfying k y a − µ a k < r the following inequality holds sup t ≥ a k y t − µ t k < ε. (B) Attractivity: µ is called attractive, if there exists r > such that for any solution y of (1.1) satisfying k y a − µ a k < r we have lim t →∞ k y t − µ t k = 0 . (C) Asymptotic stability: µ is called(i) asymptotically stable, if it is stable and attractive.(ii) exponentially stable, if it is stable and there exists r > such that for any solution y of (1.1) satisfying k y a − µ a k < r we have lim sup t →∞ t log k y t − µ t k < . ν ∈ ( , : Young systems Lemma 3.2
Let γ ( s, t ) be a control function, Λ([ s, t ]) a positive increasing function w.r.t. theinclusion of interval set [ s, t ] . Assume θ ∈ C q − var satisfying for any s, t ∈ [ a, b ] ||| θ ||| q − var , [ s,t ] ≤ γ ( s, t ) + Λ([ s, t ]) ||| x ||| p − var , [ s,t ] + 2 K Λ([ s, t ]) ||| x ||| p − var , [ s,t ] ||| θ ||| q − var , [ s,t ] . (3.1) Then for any s, t ∈ [ a, b ] ||| θ ||| q − var , [ s,t ] ≤ γ ( s, t ) + 2Λ([ s, t ]) ||| x ||| p − var , [ s,t ] + (2 K ) p − (2Λ([ s, t ])) p ||| x ||| pp − var , [ s,t ] . (3.2) Proof:
We apply the same arguments as in [17, Proposition 5.10, pp. 83-84]. Namely, for anyfixed [ s, t ] ⊂ [ a, b ], it follows from (3.1) that for [ u, v ] ⊂ [ s, t ] ||| θ ||| q − var, [ u,v ] ≤ γ ( u, v ) + 2Λ([ s, t ]) ||| x ||| p − var , [ u,v ] whenever ||| x ||| p − var , [ u,v ] ≤ K Λ([ s, t ]) . (3.3)Assume that ||| x ||| p − var , [ s,t ] > K Λ([ s,t ]) , define a sequence of greedy time t = s, t i +1 := inf { u ≥ t i , ||| x ||| p − var , [ t i ,u ] = 14 K Λ([ s, t ]) } ∧ t. The sequence would end up at some time t N = t , with( N − (cid:16) K Λ([ s, t ]) (cid:17) p = N − X i =0 ||| x ||| pp − var , [ t i ,t i +1 ] ≤ ||| x ||| pp − var , [ s,t ] , so that N − ≤ (cid:16) K Λ([ s, t ]) (cid:17) p ||| x ||| pp − var , [ s,t ] . t i , we derive k θ t − θ s k ≤ N − X i =0 k θ t i +1 − θ t i k≤ N − X i =0 (cid:16) γ ( t i , t i +1 ) + 2Λ([ s, t ]) 14 K Λ([ s, t ]) (cid:17) + 2 γ ( t N − , t N ) + 2Λ([ s, t ]) ||| x ||| p − var , [ t N − ,t N ] ≤ γ ( s, t ) + ( N −
1) 12 K + 2Λ([ s, t ]) ||| x ||| p − var , [ s,t ] ≤ γ ( s, t ) + 12 K (cid:16) K Λ([ s, t ]) (cid:17) p ||| x ||| pp − var , [ s,t ] + 2Λ([ s, t ]) ||| x ||| p − var , [ s,t ] , in case ||| x ||| p − var , [ s,t ] > K Λ([ s,t ]) . All in all, for any s, t ∈ [ a, b ] k θ t − θ s k ≤ γ ( s, t ) + 2Λ([ s, t ]) ||| x ||| p − var , [ s,t ] + (2 K ) p − (cid:16) s, t ]) (cid:17) p ||| x ||| pp − var , [ s,t ] . Using the fact that γ ( s, t ) and ||| x ||| pp − var , [ s,t ] are control functions, it follows from the definition of q -var seminorm that for all a ≤ s ≤ t ≤ b ||| θ ||| q − var , [ s,t ] ≤ γ ( s, t ) + 2Λ([ s, t ]) ||| x ||| p − var , [ s,t ] + (2 K ) p − (2Λ([ s, t ])) p ||| x ||| pp − var , [ s,t ] . Lemma 3.3
Assume that there exist positive increasing functions
H, κ , κ with Eκ ( ||| x ||| p − var , [0 , ) < ∞ ; (3.4) such that y t satisfying log k y t k ≤ log k y a k + Z ta [ H ( k y s k ) − λ A ] ds + C g κ ( ||| x ||| p − var , [ a,t ] )+ C g κ ( k y a k ) , ∀ a ≤ t ≤ a +1 . (3.5) If H (0) < λ A then there exists ǫ > such that for all C g < ǫ the zero solution is locally exponentiallystable a.s.Proof: We apply the random norm techniques in [1, Chapter 6] to translate the originalproblem for random integral inequality (3.12) into the problem for deterministic integral inequality.Fix an 0 < ǫ < λ A − H (0) − ǫEκ ( ||| x ||| p − var , [0 , ) and assignΓ( t, x ) := C g κ ( ||| x ||| p − var , [ n,t ] ) + n − X k =0 C g κ ( ||| x ||| p − var , [ k,k +1] ) , ∀ n ≥ , ∀ t ∈ [ n, n + 1] . Then it follows from (3.12) thatlog k y t k ≤ log k y n k + Z tn [ H ( k y s k ) − λ A ] ds + C g κ ( ||| x ||| p − var , [ n,t ] ) + C g κ ( k y n k ) , ∀ t ∈ [ n, n + 1] . Hence for any t ∈ [ n, n + 1]log k y t k exp { ( λ A − H (0) − ǫ ) t − Γ( t, x ) }≤ log k y n k exp { ( λ A − H (0) − ǫ ) n − Γ( n, x ) } + Z tn h H (cid:16) k y s k exp { ( λ A − H (0) − ǫ ) s − Γ( s, x ) } exp {− ( λ A − H (0) − ǫ ) s + Γ( s, x ) } (cid:17) ( H (0) + ǫ ) i ds + C g κ (cid:16) k y n k exp { ( λ A − H (0) − ǫ ) n − Γ( n, x ) } exp {− ( λ A − H (0) − ǫ ) n + Γ( n, x ) } (cid:17) . From the definitions of Γ and κ , for almost sure all x there exist the limitlim t →∞ Γ( t, x ) t = C g lim n →∞ n n − X k =0 κ ( ||| x ||| p − var , [ k,k +1] ) = C g Eκ ( ||| x ||| p − var , [0 , ) < λ A − H (0) − ǫ, (3.6)thus there exists an integer m = m ( λ A − H (0) − ǫ, x ) such that − ( λ A − H (0) − ǫ ) t + Γ( t, x ) < t ≥ m ( λ A − H (0) − ǫ, x ). Assign z t := log k y t k exp { ( λ A − H (0) − ǫ ) t − Γ( t, x ) } = log k y t k + ( λ A − H (0) − ǫ ) t − Γ( t, x ) , ∀ t ≥ . Because H and κ are increasing functions, it follows that for any n ≥ m (( λ A − H (0) − ǫ ) , x ) z t ≤ z n + C g κ ( e z n ) + Z tn h H ( e z s ) − ( H (0) + ǫ ) i ds, ∀ t ∈ [ n, n + 1] . (3.7)Again since H and κ are increasing functions, there exists a δ > C g κ ( δ ) + H ( δe C g κ ( δ ) ) < H (0) + ǫ. Using (2.4), one can choose r ( x ) such that k y k < r ( x ) = δ exp { Γ( m, x ) − ( λ A − H (0) − ǫ ) m } m − Y j =0 h { F ( ||| x ||| p − var , [ j,j +1] ) } i − , (3.8)so that (3.8) and (2.4) implies z m = log k y m k + ( λ A − H (0) − ǫ ) m − Γ( m, x ) < log δ, ∀k y k < r ( x ) . Because H (exp { z m + C g κ ( e z m ) } ) < H ( δe C g κ ( δ ) ) < H (0) + ǫ , it follows from the continuity in s of H ( e z s ) that H ( e z s ) < H (0) + ǫ, ∀ s ∈ [ m, m + τ ) for some small τ >
0. Denote by τ ∞ thesupremum of such τ and assume τ ∞ <
1, then the integral R m + τ ∞ m [ . . . ] ds in (3.7) is negative, hence z m + τ ∞ < z m + C g κ ( e z m ) < log δ + C g κ ( δ ) and H ( e z m + τ ∞ ) < H ( δe C g κ ( δ ) ) < H (0) + ǫ . Thismeans there exists τ > τ ∞ such that H ( e z s ) < H (0) + ǫ, ∀ s ∈ [ m, m + τ ) which contradicts to thedefinition of τ ∞ . Therefore τ ∞ ≥ z t < log δ + C g κ ( δ ) , ∀ t ∈ [ m, m + 1]. Again (3.7) yields z t ≤ z m + C g κ ( δ ) − h H (0) + ǫ − H (cid:16) δe C g κ ( δ ) (cid:17)i ( t − m ) , ∀ t ∈ [ m, m + 1]and in particular z m +1 ≤ z m − h H (0) + ǫ − H ( δe C g κ ( δ ) ) − C g κ ( δ ) i < z m < log δ. (3.9)By the induction principle, (3.9) holds for every n ≥ m . Then for all t ∈ [ n, n + 1] with n ≥ m , weuse (3.9) to get z t ≤ z n + C g κ ( δ ) − h H (0) + ǫ − H (cid:16) δe C g κ ( δ ) (cid:17)i ( t − n ) ≤ z m − h H (0) + ǫ − H ( δe C g κ ( δ ) ) − C g κ ( δ ) i ( n − m ) + C g κ ( δ ) − h H (0) + ǫ − H (cid:16) δe C g κ ( δ ) (cid:17)i ( t − n )12 log δ + C g κ ( δ ) − h H (0) + ǫ − H ( δe C g κ ( δ ) ) − C g κ ( δ ) i ( t − m ) . As a result,log k y t k≤ Γ( t, x ) − ( λ A − H (0) − ǫ ) t + log δ + C g κ ( δ ) − h H (0) + ǫ − H ( δe C g κ ( δ ) ) − C g κ ( δ ) i ( t − m ) ≤ Γ( t, x ) + log δ + C g κ ( δ ) + h H (0) + ǫ − H ( δe C g κ ( δ ) ) − C g κ ( δ ) i m − h λ A − H ( δe C g κ ( δ ) ) − C g κ ( δ ) i t thus lim sup t →∞ t log k y t k ≤ − h λ A − H ( δe C g κ ( δ ) ) − C g κ ( δ ) i + Eκ ( ||| x ||| p − var , [0 , ) ≤ − h H (0) + ǫ − H ( δe C g κ ( δ ) ) − C g κ ( δ ) i < . (3.10)In other words, by choosing y satisfying (3.8), the zero solution is locally exponentially stable. Lemma 3.4
Assume that there exist positive increasing functions
H, κ with Eκ ( ||| x ||| p − var , [0 , ) < ∞ ; (3.11) such that y t satisfying log k y t k ≤ log k y a k + Z ta [ H ( k y s k ) − λ A ] ds + C g κ ( ||| x ||| p − var , [ a,t ] ) , ∀ a ≤ t ≤ a + 1 . (3.12) If k H k ∞ < λ A then there exists an ǫ > such that for C g < ǫ , the zero solution is globallyexponentially stable a.s.Proof: We can choose ǫ such that given C g < ǫ < λ := λ A − C f − C g Eκ (cid:16) ||| x ||| p − var , [0 , (cid:17) . (3.13)It follows from (3.12) thatlog k y k ≤ log k y k − ( λ A − C f ) + C g κ ( ||| x ||| p − var , [0 , )or by induction for any n ∈ N log k y n k ≤ log k y k − h λ A − C f − n n − X k =0 C g κ ( ||| x ||| p − var , [ k,k +1] ) i n. (3.14)Using the ergodic Birkhorff theorem and (3.13), we then get for a.s. all realizationlim sup n →∞ log k y n k ≤ λ A − C f − C − gEκ (cid:16) ||| x ||| p − var , [0 , (cid:17) = − λ < , which proves the globally exponential stability of the zero solution. Theorem 3.5 (Local stability for Young differential equations)
Assume X · ( ω ) is a Gaus-sian process satisfying (1.4) , and ¯ ν > ν > is fixed. Assume further that conditions (1.5) , (1.6) are satisfied, where λ A > h (0) . Then the zero solution of (1.1) is locally exponentially stable foralmost sure all the trajectories x of X . If in addition λ A > C f , then we can choose ǫ so that thezero solution of (1.1) is globally exponentially stable a.s. roof: We summarize the ideas of the proof here for reader benefits. In
Step 1 we use theintegration by parts to derive the equation of log k y t k in (3.16) and the equation of θ t = y t k y t k in(3.17). The estimate of ||| θ ||| q − var , [ s,t ] is then given by (3.19) by applying Lemma 3.2. In Step 2 we derive an estimate of log k y t k in (3.21), with the help of auxilliary polinomials P i , i = 1 , . . . , λ A > C f we prove in Step 3 that e λ A − C f ) t k y t k satisfies (3.24) and (3.26), hence the globalexponential stability is followed by applying the discrete Gronwall lemma [13, Lemma 4] and choos-ing C g according to (3.28). Step 1.
As proved in [7], there exists a unique solution of (1.2) and also the backward equation.Since y ≡ y t = 0 for all t ∈ R if y = 0 (otherwise therewould be two solutions of the backward equation starting from y t and ending at zero and y , whichis a contradiction). Then observe that g ( y s ) k y s k = g ( y s ) − g (0) k y s k = R D y g ( ηy s ) y s dη k y s k = Z D y g ( ηy s ) θ s dη =: G ( y s , θ s ) , ∀ s ∈ R ; (3.15)meanwhile k f ( y s ) k = k f ( y s ) − f (0) k ≤ h ( k y s k ) k y s k , ∀ s ∈ R . Using the rule of integration by parts (see [41, 42]), it is easy to check that d log k y t k = h θ t , Aθ t + f ( y t ) k y t k i dt + h θ t , G ( y t , θ t ) i dx t , (3.16)where θ t satisfies the equation dθ t = (cid:16) Aθ t + f ( y t ) k y t k − θ t h θ t , Aθ t + f ( y t ) k y t k i (cid:17) dt + (cid:16) G ( y t , θ t ) − θ t h θ t , G ( y t , θ t ) i (cid:17) dx t . (3.17)A direct computation using assumptions shows that k G ( y, θ ) k ∞ , [ a,b ] ≤ C g and ||| G ( y, θ ) ||| q − var , [ a,b ] = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z D y g ( ηy ) θdη (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q − var , [ a,b ] ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z D y g ( ηy ) dη (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q − var , [ a,b ] k θ k ∞ ,a,b + (cid:13)(cid:13)(cid:13)(cid:13)Z D y g ( ηy ) dη (cid:13)(cid:13)(cid:13)(cid:13) ∞ , [ a,b ] ||| θ ||| q − var , [ a,b ] ≤ C g (cid:16) ||| θ ||| q − var , [ a,b ] + 12 ||| y ||| q − var , [ a,b ] (cid:17) . (3.18)it follows that k θ t − θ s k ≤ k A k ( t − s ) + 2 Z ts h ( k y u k ) du + 2 C g ||| x ||| p − var , [ s,t ] + K ||| x ||| p − var , [ a,t ] ||| G ( y, θ ) − θ h θ, G ( y, θ ) i||| q − var , [ s,t ] ≤ k A k ( t − s ) + 2 Z ts h ( k y u k ) du + 2 C g ||| x ||| p − var , [ s,t ] + KC g ||| x ||| p − var , [ s,t ] ||| y ||| q − var , [ s,t ] + 4 KC g ||| x ||| p − var , [ s,t ] ||| θ ||| q − var , [ s,t ] . Since each of t − s, R ts h ( k y u k ) du, ||| x ||| p − var , [ s,t ] ||| x ||| q − var , [ s,t ] is a control, the function γ ( s, t ) := 2 k A k ( t − s ) + 2 Z ts h ( k y u k ) du + KC g ||| x ||| p − var , [ s,t ] ||| y ||| q − var , [ s,t ]
14s also a control. By using triangle inequality for q -var seminorm with q ≥ p ≥
1, we get for all a ≤ s < t ≤ b k θ t − θ s k ≤ ||| θ ||| q − var , [ s,t ] = sup Π n X [ u,v ] ∈ Π (cid:16) γ ( u, v ) + 2 C g ||| x ||| p − var , [ u,v ] + 4 KC g ||| x ||| p − var , [ s,t ] ||| θ ||| q − var , [ u,v ] (cid:17) q o q ≤ γ ( s, t ) + 2 C g ||| x ||| p − var , [ s,t ] + 4 KC g ||| x ||| p − var , [ s,t ] ||| θ ||| q − var , [ s,t ] , which has the form of (3.1) with Λ([ s, t ]) := 2 C g . Applying (3.2) in Lemma 3.2 we conclude thatfor all a ≤ s ≤ t ≤ b ||| θ ||| q − var , [ s,t ] ≤ γ ( s, t ) + 4 C g ||| x ||| p − var , [ s,t ] + (2 K ) p − (4 C g ) p ||| x ||| pp − var , [ s,t ] ≤ k A k ( t − s ) + 4 Z ts h ( k y u k ) du + 2 KC g ||| x ||| p − var , [ s,t ] ||| y ||| q − var , [ s,t ] +4 C g ||| x ||| p − var , [ s,t ] + (2 K ) p − (4 C g ) p ||| x ||| pp − var , [ s,t ] . (3.19) Step 2.
Next, to estimate (3.16), we first use (2.2) and (3.18) to get (cid:13)(cid:13)(cid:13)(cid:13)Z ba h θ s , G ( y s , θ s ) i dx s (cid:13)(cid:13)(cid:13)(cid:13) ≤ C g ||| x ||| p − var , [ a,b ] + K ||| x ||| p − var , [ a,b ] |||h θ, G ( y, θ ) i||| q − var , [ a,b ] ≤ ||| x ||| p − var , [ a,b ] (cid:16) C g + 2 KC g ||| θ ||| q − var, [ a,b ] + 12 KC g ||| y ||| q − var , [ a,b ] (cid:17) . We estimate equation (3.16) in the integration form, using (3.19) and (1.5)log k y t k ≤ log k y a k + Z ta [ − λ A + h ( k y s k )] ds + ||| x ||| p − var , [ a,t ] (cid:16) C g + 2 KC g ||| θ ||| q − var, [ a,b ] + 12 KC g ||| y ||| q − var , [ a,b ] (cid:17) ≤ log k y a k + Z ta [ − λ A + h ( k y s k )] ds + C g ||| x ||| p − var , [ a,t ] + 12 KC g ||| x ||| p − var , [ a,t ] ||| y ||| q − var , [ a,t ] +2 KC g ||| x ||| p − var , [ a,t ] n k A k + C f )( t − a ) + 2 KC g ||| x ||| p − var , [ a,t ] ||| y ||| q − var , [ a,t ] +4 C g ||| x ||| p − var , [ a,t ] + (2 K ) p − (4 C g ) p ||| x ||| pp − var , [ a,t ] o . Writing in short Y a,t := ||| y ||| q − var , [ a,t ] and x p = ||| x ||| p − var , [ a,t ] , we then get for all 0 ≤ a < t ≤ a + 1log k y t k ≤ log k y a k + Z ta [ − λ A + h ( k y s k )] ds + C g x p + 12 KC g x p Y a,t +2 KC g x p n k A k + C f )( t − a ) + 2 KC g x p Y a,t + 4 C g x p + (2 K ) p − (4 C g x p ) p o ≤ log k y a k + Z ta [ − λ A + h ( k y s k )] ds + (cid:16) KC g x p + 4 K C g x p (cid:17) Y a,t + C g h x p + 8 K ( k A k + C f ) x p + 8 KC g x p + (8 KC g ) p x p +1 p i . (3.20)On the other hand, it follows from (2.4) and Cauchy inequality that (cid:16) KC g x p + 4 K C g x p (cid:17) Y a,t ≤ (cid:16) Kx p + 4 K C g x p (cid:17) C g k y a k exp n F ( x p ) o ≤ C g k y a k + 12 C g (cid:16) Kx p + 4 K C g x p (cid:17) exp n F ( x p ) o .
15n summary, we have just proved that for all a ≤ t ≤ a + 1log k y t k ≤ log k y a k + Z ta h − λ A + h ( k y s k ) i ds + C g P ( x p , e x p ) + 12 C g k y a k (3.21)where P ( x , x ) is a polynomial with positive coefficients depending on C g such that P (0 , x ) = P ( x ,
0) = 0 . (3.22)Assign H ( z ) := h ( z ) , κ ( z ) := P ( z, e z ) , κ ( z ) := 12 z . (3.23)Since the random variable Z := e ||| x ||| p − var , [0 , has finite moments of any order for 1 < p < x to be a realization of Gaussian stochastic process, it follows that κ satisfies (3.11). Hence using λ A > h (0) the conclusion of local stability is therefore a direct consequence of Lemma 3.3. Step 3.
Assume λ A > C f and assign λ := λ A − C f >
0, then we apply the integration by partsto get de λt k y t k = 2 λe λt k y t k dt + 2 e λt h y t , Ay t + f ( y t ) i dt + 2 e λt h y t , g ( y t ) i dx t , or in the integral form e λt k y t k = k y k + 2 Z t e λs (cid:16) λ k y s k + h y s , Ay s + f ( y s ) i (cid:17) ds + 2 Z t e λs h y s , g ( y s ) i dx s . (3.24)Using (1.5), the first integral in (3.24) is then non-positive, thus for any n ∈ N e λn k y n k ≤ k y k + n − X k =0 (cid:13)(cid:13)(cid:13) Z k +1 k e λs h y s , g ( y s ) i dx s (cid:13)(cid:13)(cid:13) ≤ k y k + n − X k =0 ||| x ||| p − var , [ k,k +1] (cid:16) e λk kh y k , g ( y k ) ik + K (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e λ · h y, g ( y ) i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q − var , [ k,k +1] (cid:17) . (3.25)Observe that kh y s , g ( y s ) k ≤ C g k y s k and due to (2.4) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e λ · h y, g ( y ) i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q − var , [ s,t ] ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e λ · (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q − var , [ s,t ] C g k y k ∞ , [ s,t ] + 2 e λt C g k y k ∞ , [ s,t ] ||| y ||| q − var , [ s,t ] ≤ (cid:16) e λt − e λs (cid:17) C g h { F ( ||| x ||| q − var , [ s,t ] ) } i k y s k +2 e λt C g h { F ( ||| x ||| q − var , [ s,t ] ) } i exp { F ( ||| x ||| q − var , [ s,t ] ) }k y s k ≤ C g e λs k y s k n(cid:16) e λ ( t − s ) − (cid:17)h { F ( ||| x ||| q − var , [ s,t ] ) } i +2 e λ ( t − s ) h { F ( ||| x ||| q − var , [ s,t ] ) } i exp { F ( ||| x ||| q − var , [ s,t ] ) } o ≤ C g e λs k y s k κ ( t − s, ||| x ||| q − var , [ s,t ] ) , where κ ( u, v ) := ( e λu − e F ( v ) ] + 2 e λu [1 + e F ( v ) ] e F ( v ) . Hence it follows from (3.25) that e λn k y n k ≤ k y k + n − X k =0 C g ||| x ||| p − var , [ k,k +1] h κ (1 , ||| x ||| p − var , [ k,k +1] ) + 1 i e λk k y k k . (3.26)16pplying the discrete Gronwall lemma in [13, Lemma 4] for the sequence e λn k y n k with parameters2 C g ||| x ||| p − var , [ k,k +1] h κ (1 , ||| x ||| p − var , [ k,k +1] ) + 1 i in (3.26), we get e λn k y n k ≤ k y k n − Y k =0 (cid:16) C g ||| x ||| p − var , [ k,k +1] h κ (1 , ||| x ||| p − var , [ k,k +1] ) + 1 i(cid:17) . Taking the logarithm on both sides, then dividing by 2 n and letting n tend to infinity we get, dueto the inequality log(1 + r ) ≤ r, ∀ r > n →∞ n log k y n k ≤ − λ + lim n →∞ n n − X k =0 log (cid:16) C g ||| x ||| p − var , [ k,k +1] h κ (1 , ||| x ||| p − var , [ k,k +1] ) + 1 i(cid:17) ≤ − λ + lim n →∞ n n − X k =0 C g ||| x ||| p − var , [ k,k +1] h κ (1 , ||| x ||| p − var , [ k,k +1] ) + 1 i ≤ − λ + C g E ||| x ||| p − var , [0 , h κ (1 , ||| x ||| p − var , [0 , ) + 1 i , where the expectation E ||| x ||| p − var , [0 , h κ (1 , ||| x ||| p − var , [0 , )+1 i is finite due to the fact that Ee Λ F ( ||| x ||| p − var , [0 , ) is finite for any Λ >
0. Finally, for any t ∈ [ n, n + 1], we use (2.4) to getlim sup t →∞ t log k y t k ≤ lim sup n →∞ n log k y n k + lim sup n →∞ n log[1 + exp { F ( ||| x ||| p − var , [ n,n +1] ) } ] ≤ − λ + C g E ||| x ||| p − var , [0 , h κ (1 , ||| x ||| p − var , [0 , ) + 1 i + lim sup n →∞ n [1 + F ( ||| x ||| p − var , [ n,n +1] )] ≤ − λ + C g E ||| x ||| p − var , [0 , h κ (1 , ||| x ||| p − var , [0 , ) + 1 i , (3.27)where the second limsup in the right hand side of (3.27) is zero due to the integrability of F ( ||| x ||| p − var , [0 , ).Hence if we choose C g < ǫ < ( λ A − C f ) 1 E ||| x ||| p − var , [0 , h κ (1 , ||| x ||| p − var , [0 , ) + 1 i , (3.28)then the zero solution is globally exponentially stable a.s. Corollary 3.6
Assume that the linear Young differential equation dy t = Ay t dt + Cy t dx (3.29) satisfies (1.5) . Then a criterion for the globally exponential stability is λ A > (cid:16) K k A k + 8 K + (8 K ) p (cid:17) k C k (cid:16) E ||| x ||| p +1 p − var , [0 , (cid:17) p +1 (3.30) Proof:
For h ( · ) ≡ , g ( y ) = Cy , there is no term Y a,t in (3.20), hence it follows from (3.21)thatlog k y k +1 k ≤ log k y k k − λ A + (1 + 4 K k A k ) k C k ||| x ||| p − var , [ k,k +1] + 8 K k C k ||| x ||| p − var , [ k,k +1] +(8 K ) p k C k p +1 ||| x ||| p +1 p − var , [ k,k +1] .
17s a resultlim sup n →∞ n log k y n k ≤ − λ A + (1 + 4 K k A k ) k C k E ||| x ||| p − var , [0 , + 8 K k C k E ||| x ||| p − var , [0 , +(8 K ) p k C k p +1 E ||| x ||| p +1 p − var , [0 , , ≤ − λ A + (1 + 4 K k A k ) k C k (cid:16) E ||| x ||| p +1 p − var , [0 , (cid:17) p +1 + 8 K k C k (cid:16) E ||| x ||| p +1 p − var , [0 , (cid:17) p +1 +(8 K ) p k C k p +1 E ||| x ||| p +1 p − var , [0 , . Assign ˜ C := k C k (cid:16) E ||| x ||| p +1 p − var , [0 , (cid:17) p +1 , then system (3.29) is exponentially stable if λ A > (1 + 4 K k A k ) ˜ C + 8 K ˜ C + (8 K ) p ˜ C p +1 , (3.31)which, together with the fact that λ A < k A k and K >
1, implies that ˜
C <
1. In that case (3.31) isfollowed from (3.30). ν ∈ ( , ) and g ( y ) = Cy In this section we consider a particular rough case in which g ( y ) = Cy . We could then prove thesame conclusions on stability, and even a general form of local stability. Theorem 3.7 (Local stability for rough differential equations)
Assume > ¯ ν > ν > and X · ( ω ) is a stationary process satisfying (1.4) . Assume further that conditions (1.5) , (1.6) aresatisfied, where g ( y ) = Cy and λ > h (0) . Then there exists an ǫ > such that given k C k < ǫ , thezero solution of (1.1) is locally exponentially stable for almost all realization x of X . If in addition λ > C f , then we can choose ǫ so that the zero solution of (1.1) is globally exponentially stable.Proof: We sketch out the proof here in several steps. In
Step 1 , we derive the equation forlog k y t k in (3.32), and the equation for θ = y k y k in (3.33). Notice that for Gaussian geometric roughpath, then [ x ] · , · = 0, but we still compute the estimates here for general rough paths. As such theestimate for ||| θ, θ ′ ||| x, α, [ a,b ] is proved by Proposition 3.8 which, due to G ( y ) = Cy , does not include ||| y, y ′ ||| x, α, [ a,b ] , hence we do not need the integrability of ||| y, y ′ ||| x, α, [ a,b ] . The estimate for log k y t k isthen derived in (3.37) in Step 2 , where each component is computed so that finally log k y t k satisfies(3.41). The conclusion is then followed from Proposition 3.8 and Theorem 3.5. Step 1.
We use similar arguments in [14] to prove that the solution of the pathwise solutionof the linear rough differential equation (1.1) generates a linear rough flow on R d , and that y t = 0iff y = 0. Hence it remains to prove all the formula for θ t and r t . By direct computations using(2.10), we can show the following equations. • k y t k satisfies the RDE d k y t k = 2 h y t , Ay t + f ( y t ) i dt + 2 h y t , Cy t i dx t + k Cy t k d [ x ] ,t , where 2 h y, Cy i ′ s = 2 h y ′ s , Cy s i + 2 h y s , [ Cy ] ′ s i . • k y t k satisfies the RDE d k y t k = 1 k y t k h y t , Ay t + f ( y t ) i dt + 1 k y t k h y t , Cy t i dx t + 12 k y t k h k Cy t k − k y t k h y t , Cy t i i d [ x ] ,t , where h k y k h y, Cy i i ′ s = h k y k i ′ s h y s , Cy s i + k y s k h h y, Cy i i ′ s .18 log k y t k satisfies the RDE d log k y t k = h θ t , Aθ t + f ( y t ) k y t k i dt + h θ t , Cθ t i dx t + h k Cθ t k − h θ t , Cθ t i i d [ x ] ,t , (3.32)where h h θ, Cθ i i ′ s = h θ ′ s , Cθ s i + h θ s , [ Cθ ] ′ s i . • θ t satisfies the RDE dθ t = h Aθ t − h θ t , Aθ t i θ t + f ( y t ) k y t k − h θ t , f ( y t ) k y t k i θ t i dt + h Cθ t − h θ t , Cθ t i θ t i dx t + 12 n h θ t , Cθ t i θ t − h θ t , Cθ t i Cθ t − k Cθ t k θ t o d [ x ] ,t , (3.33)where h Cθ − h θ, Cθ i θ i ′ s = [ Cθ ] ′ s − h θ s , Cθ s i θ ′ s − h h θ s , Cθ s i i ′ s θ s . Rewrite (3.33) in the form dθ t = f ( t, θ t ) dt + g ( θ t ) dx t + k ( θ t ) d [ x ] a,t , t ∈ [ a, b ] (3.34)or in the integral form θ t = F ( θ, θ ′ ) t = θ a + Z ta f ( u, θ u ) du + Z ta g ( θ u ) dx u + Z ta k ( θ u ) d [ x ] a,u , ∀ ≤ a ≤ t ≤ b ;where g ∈ C such that there exist C g := max n k g ( θ ) k ∞ , [0 ,T ] , k D θ g ( θ ) k ∞ , [0 ,T ] , k D θθ g ( θ ) k ∞ , [0 ,T ] o < ∞ ; k is Lipschitz continuous with Lipschitz constant such that C k := k k ( θ ) k ∞ , [0 ,T ] ∨ Lip( k ) < ∞ . We can prove the following estimate (see the proof in the Appendix).
Proposition 3.8
There exist a generic constant P = P ( b − a, ν − α ) such that for all ≤ a ≤ b ≤ a + 1 , max n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( θ, θ ′ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x, α, [ a,b ] , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( θ, θ ′ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x, α, [ a,b ] , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( θ, θ ′ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x, α, [ a,b ] o ≤ P ( b − a ) M ν − α h (cid:16) ||| x ||| ν, [ a,b ] + ||| X ||| ν, ∆ ([ a,b ]) + ||| [ x ] ||| ν, ∆([ a,b ]) (cid:17) ν − α i (3.35) where M := max n C f , C g (1 + C α ) , C k (1 + K α ) , C g ( C α + 1) , o (3.36) Step 2.
It is now sufficient to estimate the quantity in (3.32). For any 0 ≤ a ≤ t ≤
1, rewrite(3.32) in the integral formlog k y t k = log k y a k + Z ta h y s , Ay s + f ( y s ) k y s k i ds + Z ta h θ s , Cθ s i dx s + Z ta h k Cθ s k − h θ s , Cθ s i i d [ x ] ,s ≤ log k y a k − λ ( t − a ) + Z ta h ( k y s k ) ds + (cid:13)(cid:13)(cid:13) Z ta h θ s , Cθ s i dx s (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13) Z ta h k Cθ s k − h θ s , Cθ s i i d [ x ] a,s (cid:13)(cid:13)(cid:13) . (3.37)The last term in the last line of (3.37) can be estimated as (cid:13)(cid:13)(cid:13) Z ba h k Cθ s k − h θ s , Cθ s i i d [ x ] a,s (cid:13)(cid:13)(cid:13) ≤ k C k (cid:12)(cid:12)(cid:12) [ x ] a,b (cid:12)(cid:12)(cid:12) + K α | b − a | α ||| [ x ] ||| α, ∆ ([ a,b ]) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)h k Cθ k − h θ, Cθ i i(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α, [ a,b ] ≤ k C k | b − a | α ||| [ x ] ||| α, ∆ ([ a,b ]) + K α | b − a | α ||| [ x ] ||| α, ∆ ([ a,b ]) h k C k + 4 k C k i ||| θ ||| α, [ a,b ] ≤ k C k | b − a | α ||| [ x ] ||| α, ∆ ([ a,b ]) h
32 + 5 K α | b − a | α (cid:16) C g ||| x ||| α + | b − a | ν − α ( ||| x ||| α + 1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) θ, θ ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x, α (cid:17)i ≤ k C k | b − a | α ||| [ x ] ||| α, ∆ ([ a,b ]) (cid:16)
32 + 5 K α C G | b − a | α ||| x ||| α (cid:17) +5 K α k C k | b − a | h ||| [ x ] ||| α, ∆ ([ a,b ]) ( ||| x ||| α + 1) + 12 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) θ, θ ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x, α i . (3.38)Meanwhile the rough integral can be estimated as (cid:13)(cid:13)(cid:13) Z ba h θ s , Cθ s i dx s (cid:13)(cid:13)(cid:13) ≤ (cid:12)(cid:12)(cid:12) h θ a , Cθ a i (cid:12)(cid:12)(cid:12) | x b − x a | + (cid:12)(cid:12)(cid:12) h θ, Cθ i ′ a (cid:12)(cid:12)(cid:12) | X a,b | + C α | b − a | α (cid:16) ||| x ||| α, [ a,b ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R h θ,Cθ i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α, [ a,b ] + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h θ, Cθ i ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α, [ a,b ] ||| X ||| α, ∆ ([ a,b ]) (cid:17) ≤ k C k| b − a | α ||| x ||| α, [ a,b ] + 4 k C k | b − a | α ||| X ||| α, ∆ ([ a,b ]) + C α | b − a | α (cid:16) ||| x ||| α, [ a,b ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R h θ,Cθ i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α, [ a,b ] + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h θ, Cθ i ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α, [ a,b ] ||| X ||| α, ∆ ([ a,b ]) (cid:17) . (3.39)To estimate the brackets of the last line of (3.39), we apply (4.1) to get (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h θ, Cθ i ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α, [ a,b ] ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k Cθ k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α, [ a,b ] + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h θ, C θ i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α, [ a,b ] + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h θ, Cθ i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α, [ a,b ] + |||h θ, Cθ ih θ, Cθ i||| α, [ a,b ] ≤ k C k ||| θ ||| α, [ a,b ] ≤ k C k (cid:16) C g ||| x ||| α + | b − a | ν − α ( ||| x ||| α + 1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) θ, θ ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x, α (cid:17) . Meanwhile k R h θ,Cθ i s,t k ≤ (cid:12)(cid:12)(cid:12) h θ t , Cθ t i − h θ s , Cθ s i − h θ · , Cθ i ′ s x s,t (cid:12)(cid:12)(cid:12) ≤ k C kk R θs,t k + 2 k C kk θ ′ s kk R θs,t kk x s,t k + k C kk R θs,t k + k C kk θ ′ s k k x s,t k ≤ k C kk R θs,t k + 4 k C k k R θs,t kk x s,t k + k C kk R θs,t k + 4 k C k k x s,t k ;thus it follows that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R h θ,Cθ i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α, [ a,b ] ≤ k C k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R θ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α, [ a,b ] + 4 k C k | b − a | α ||| x ||| α, [ a,b ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R θ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α, [ a,b ] + k C k| b − a | α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R θ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α, [ a,b ] +4 k C k ||| x ||| α, [ a,b ] ≤ k C k ||| x ||| α, [ a,b ] + (cid:16) k C k + 4 k C k | b − a | α ||| x ||| α, [ a,b ] (cid:17) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) θ, θ ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x, α + k C k| b − a | α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) θ, θ ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x, α . (cid:13)(cid:13)(cid:13) Z ba h θ s , Cθ s i dx s (cid:13)(cid:13)(cid:13) ≤ k C k| b − a | α ||| x ||| α, [ a,b ] + 4 k C k | b − a | α ||| X ||| α, ∆ ([ a,b ]) + C α k C k | b − a | α (cid:16) k C k ||| x ||| α, [ a,b ] ||| x ||| α, [ a,b ] + 14 C G ||| X ||| α, ∆ ([ a,b ]) ||| x ||| α, [ a,b ] (cid:17) + C α k C k| b − a | α n ||| x ||| α, [ a,b ] (cid:16) k C k| b − a | α ||| x ||| α, [ a,b ] (cid:17) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) θ, θ ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x, α + | b − a | α (cid:16) ||| x ||| α, [ a,b ] + 12 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) θ, θ ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x, α (cid:17) +7 k C k| b − a | ν − α h ( ||| x ||| α + 1) ||| X ||| α, [ a,b ] + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) θ, θ ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x, α io . (3.40)Replacing (3.38) and (3.40) into (3.37) using (3.35) in Lemma 3.8, we conclude that there exists anincreasing polynomial with all positive coefficients κ ( t, a, x, X , [ x ]) = κ (cid:16) t − a, ||| x ||| α, [ a,t ] , ||| X ||| α, ∆ ([ a,t ]) , ||| [ x ] ||| α, ∆ ([ a,t ]) (cid:17) , κ ( a, a, x, X , [ x ]) = 0 , and an increasing function K : R + → R + such that for all 0 ≤ a ≤ t ≤ k y t k ≤ log k y a k + Z ta h h ( k y s k ) + k C k K ( k y s k ) − λ A i ds + k C k κ ( t, a, x, X , [ x ]) , (3.41)which is similar to (3.37). Because of (2.7), (3.6) holds for the realization x and X . Since λ > k (0),we can choose k C k < ǫ small enough such that function H ( u ) := h ( u ) + k C k K ( u ) is increasingfunction and H (0) < λ A . Using (2.17), Lemma 3.3 and Lemma 3.4, we can then prove that system(1.1) is locally/globally exponentially stable at zero for almost sure all the realization. Corollary 3.9
Let Φ( t, x, X , [ x ]) be the solution matrix of dz t = Az t dt + Cz t dx t . Then there existsa function κ ( t, a, x, X , [ x ]) such that for any δ > k Φ( t, x, X , [ x ]) k ≤ exp n − λ A t + k C k κ ( t, , x, X , [ x ]) o . (3.42) As a result lim sup t →∞ t log k z t k ≤ − λ A + k C k E κ ( δ, , x, X , [ x ]) . (3.43) Corollary 3.10
Consider the following system dy t = [ Ay t + f ( y t )] dt + Cy t dB Ht , y · ∈ R d , (3.44) where B H is a fractional Brownian motion with Hurst index < H < ; A is negative definite and f : R d → R d is globally Lipschitz continuous, i.e. there exist contants h , c f > such that h y, Ay i ≤ − h k y k , k f ( y ) − f ( y ) k ≤ c f k y − y k , ∀ y , y ∈ R d . (3.45) Assume that h > c f . There exists an ǫ > such that under condition k C k < ǫ , ϕ possesses arandom pullback attractor consisting only one point a ( x ) , to which other random points converge towith exponential rate.Proof: The case
H > is proved in [14, Theorem 3.3]. For < H < , starting with theestimate (3.42), we apply the H¨older inequality such that κ ( t, a, x, X , [ x ]) ≤ H + ( t − a )˜ κ ( t, a, x, X , [ x ]) , ∀ ≤ a ≤ t ≤ , H > κ ( t, a, x, X , [ x ]) = ˜ κ (cid:16) t − a, ||| x ||| α, [ a,t ] , ||| [ x ] ||| α, ∆ ([ a,t ]) , ||| X ||| α, ∆ ([ a,t ]) (cid:17) , ˜ κ ( a, a, x, X ) = 0 , and ˜ κ is an increasing function. It follows that Γ( t, s, x, X , [ x ]) = ( t − s )˜ κ ( t, s, x, X , [ x ]) is a controlfunction, and k Φ( t, x, X , [ x ]) k ≤ exp n k C k H − λ A t + k C k Γ( t, , x, X , [ x ]) o . The arguments are then similar to the proof of [14, Theorem 4.4]. We stress here that for the roughcase, it is proved in [2] that the system (3.44) generates a random dynamical system [1].
Proof: [ Proposition 3.8 ] Consider the solution mapping M : D αx ( θ a , g ( θ a )) → D αx ( θ a , g ( θ a ))defined by M ( θ, θ ′ ) t = ( F ( θ, θ ′ ) t , g ( θ t )) , together with the seminorm (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( θ, θ ′ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x, α = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) θ ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R θ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M ( θ, θ ′ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x, α = ||| g ( θ ) ||| α + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R F ( θ,θ ′ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α . We are going to estimate these seminorms. Observe from (3.34) that θ ′ = g ( θ t ), thus ||| θ ||| α ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) θ ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ ||| x ||| α + | T − a | α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R θ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α ≤ C g ||| x ||| α + | b − a | α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) θ, θ ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x, α ; (4.1) ||| g ( θ ) ||| α ≤ ||| D θ g ( θ · ) ||| ∞ ||| θ ||| α ≤ C g ||| θ ||| α . (4.2)Meanwhile using H¨older inequality k R F ( θ,θ ′ ) s,t k ≤ Z ts k f ( u, θ u ) k du + k D θ g ( θ s ) g ( θ s ) k| X s,t | + k k ( θ · ) k ∞ | [ x ] s,t | + K α | t − s | α ||| k ( θ ) ||| α ||| [ x ] ||| α + C α | t − s | α (cid:16) ||| x ||| α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R g ( θ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) g ( θ ) ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α ||| X ||| α (cid:17) ≤ | t − s | ν (cid:16) Z ba k f ( u, θ u ) k − ν du (cid:17) − ν + C g | X s,t | + C k | [ x ] s,t | + K α | t − s | α C k ||| θ ||| α ||| [ x ] ||| α + C α | t − s | α (cid:16) ||| x ||| α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R g ( θ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) g ( θ ) ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α ||| X ||| α (cid:17) , (4.3)where we use the fact that θ ′ = g ( θ ) to get (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) g ( θ ) ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) D θ g ( θ ) θ ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α ≤ ||| D θ g ( θ ) ||| ∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) θ ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α + ||| D θ g ( θ ) ||| α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) θ ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ ≤ C g ||| g ( θ ) ||| α + C g ||| θ ||| α k g ( θ ) k ∞ ≤ C g ||| θ ||| α . On the other hand k R g ( θ ) s,t k ≤ Z (cid:13)(cid:13)(cid:13) D θ g (cid:16) θ s + η ( θ t − θ s ) (cid:17) − D θ g ( θ s ) (cid:13)(cid:13)(cid:13) k θ ′ s k| x ( t ) − x ( s ) | dη + Z (cid:13)(cid:13)(cid:13) D θ g (cid:16) θ s + η ( θ t − θ s ) (cid:17)(cid:13)(cid:13)(cid:13) dη k R θs,t k , thus (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R g ( θ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α ≤ k D θ g ( θ ) k ∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R θ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α + 12 C g ||| g ( θ s ) ||| ∞ ||| x ||| α ||| θ ||| α ≤ C g (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R θ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α + 12 C g ||| x ||| α ||| θ ||| α . a < b ≤ a + 1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R F ( θ,θ ′ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α ≤ ( b − a ) ν − α (cid:16) Z ba k f ( u, θ u ) k − ν du (cid:17) − ν + C g ||| X ||| α + C k ||| [ x ] ||| α + K α C k | b − a | α ||| θ ||| α ||| [ x ] ||| α + C α | b − a | α n ||| x ||| α h C g (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R θ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α + 12 C g ||| x ||| α ||| θ ||| α i + 2 C g ||| θ ||| α ||| X ||| α o ≤ C f ( b − a ) ν − α + C g ||| X ||| α + C k ||| [ x ] ||| α + C α C g ( b − a ) α ||| x ||| α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R θ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α + n C g C α ||| X ||| α + C k K α ||| [ x ] ||| α + 12 C α C g ||| x ||| α o | b − a | α ||| θ ||| α . Together with (4.1) and (4.2) we conclude that for any a < b such that b − a ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R F ( θ,θ ′ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α + ||| g ( θ ) ||| α ≤ C f ( b − a ) ν − α + C g ||| X ||| α + C k ||| [ x ] ||| α + C α C g ( b − a ) α ||| x ||| α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) θ, θ ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x, α + nh C g C α ||| X ||| α + C k K α ||| [ x ] ||| α + 12 C α C g ||| x ||| α i | b − a | α + C g o ×× h C g ||| x ||| α + | b − a | α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) θ, θ ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x, α i ≤ C f ( b − a ) ν − α + C g ||| X ||| α + C k ||| [ x ] ||| α + C g ||| x ||| α + h C g C α ||| X ||| α + C k K α ||| [ x ] ||| α + 12 C α C g ||| x ||| α i C g | b − a | α ||| x ||| α + nh C g C α ||| X ||| α + C k K α ||| [ x ] ||| α + 12 C α C g ||| x ||| α i ( b − a ) α + C g + C α C g ||| x ||| α o ×× ( b − a ) α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) θ, θ ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x, α ≤ M h | b − a | ν − α + ||| X ||| α + ||| [ x ] ||| α + ||| x ||| α + (cid:16) ||| X ||| α + ||| x ||| α + ||| [ x ] ||| α (cid:17) ( b − a ) α ||| x ||| α i + M n(cid:16) ||| X ||| α + ||| x ||| α + ||| [ x ] ||| α (cid:17) ||| x ||| α + | b − a | ν − α + ||| X ||| α + ||| [ x ] ||| α + ||| x ||| α o (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) θ, θ ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x, α . Now construct for any fixed µ ∈ (0 ,
1) a sequence of stopping times { τ k } k ∈ N such that τ = 0 and | τ k +1 − τ k | ν − α + | τ k +1 − τ k | ν − α (cid:16) ||| x ||| ν, [ τ k ,τ k +1 ] + ||| X ||| ν, ∆ ([ τ k ,τ k +1 ]) + ||| [ x ] ||| ν, ∆ ([ τ k ,τ k +1 ]) (cid:17) = µ M , (4.4)for all k ∈ N , then it follows that ||| x ||| α ≤ | τ k +1 − τ k | ν − α ||| x ||| ν, [ τ k ,τ k +1 ] < , ||| X ||| α ≤ | τ k +1 − τ k | ν − α ) ||| X ||| ν, ∆ ([ τ k ,τ k +1 ]) < , ||| [ x ] ||| α ≤ | τ k +1 − τ k | ν − α ) ||| [ x ] ||| ν, ∆ ([ τ k ,τ k +1 ]) < , hence it derives ||| g ( θ ) ||| α, [ τ k ,τ k +1 ] + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R F ( θ,θ ′ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α, [ τ k ,τ k +1 ] ≤ M n | τ k +1 − τ k | ν − α + | τ k +1 − τ k | ν − α (cid:16) ||| x ||| ν + ||| X ||| ν + ||| [ x ] ||| ν (cid:17)o (1 + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) θ, θ ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x, α ) ≤ µ + µ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) θ, θ ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x, α . Hence using the fact that θ ′ = g ( θ ) and F ( θ, θ ′ ) = θ we conclude that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( θ, θ ′ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x, α, [ τ k ,τ k +1 ] ≤ µ − µ . (4.5)23herefore (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( θ, θ ′ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x, α, [ a,b ] ≤ µ − µ N µ M , [ a,b ] ,ν,α ( x ) , where N µ M , [ a,b ] ,ν,α ( x ) is the number of stopping times τ k in the interval [ a, b ]. It is easy to see that b − a > N µ M , [ a,b ] ,ν,α ( x ) n µ M (cid:16) ||| x ||| ν, [ a,b ] + ||| X ||| ν, ∆ ([ a,b ]) + ||| [ x ] ||| ν, ∆ ([ a,b ]) (cid:17) − o ν − α . All in all, we have just shown that for all 0 ≤ a ≤ b ≤ T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( θ, θ ′ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x, α, [ a,b ] ≤ b − a (1 − µ ) µ ν − α − (2 M ) ν − α (cid:16) ||| x ||| ν, [ a,b ] + ||| X ||| ν, ∆ ([ a,b ]) + ||| [ x ] ||| ν, ∆ ([ a,b ]) (cid:17) ν − α ≤ ( b − a ) µ − µ ) (cid:16) Mµ (cid:17) ν − α h (cid:16) ||| x ||| ν, [ a,b ] + ||| X ||| ν, ∆ ([ a,b ]) + ||| [ x ] ||| ν, ∆ ([ a,b ]) (cid:17) ν − α i . (4.6)The other estimates for ||| ( θ, θ ′ ) ||| x, α, [ a,b ] and ||| ( θ, θ ′ ) ||| x, α, [ a,b ] are direct consequences of Cauchyinequality for (4.6). Acknowledgments
This work was supported by the Max Planck Institute for Mathematics in the Science (MIS-Leipzig).
References [1] L. Arnold.
Random Dynamical Systems.
Springer, Berlin Heidelberg New York, 1998.[2] I. Bailleul, S. Riedel, M. Scheutzow.
Random dynamical systems, rough paths and rough flows.
J. Differential Equations, Vol. , (2017), 5792–5823.[3] T. A. Burton.
Volterra integral and differential equations.
Mathematics in Science and Engi-neering, Edited by C.K. Chui, Stanford University, Vol. , 2005.[4] T. Cass, P. Friz.
Densities for rough differential equations under H¨ormander conditions.
Annalsof Mathematics, Vol. , (2010), 2115–2141.[5] T. Cass, C. Litterer, T. Lyons.
Integrability and tail estimates for Gaussian rough differentialequations.
Annals of Probability, Vol. , No. 4, (2013), 3026–3050.[6] T. Cass, M. Hairer, C. Litterer and S. Tindel. Smoothness of the density for solutions to Gaussianrough differential equations.
The Annals of Probability, Vol. , No. 1, (2015), 188–239.[7] N. D. Cong, L. H. Duc, P. T. Hong. Young differential equations revisited.
J. Dyn. Diff. Equat.,Vol. , Iss. , (2018), 1921–1943.[8] L.Coutin. Rough paths via sewing lemma.
ESAIM: Probability and Statistics., , (2012),479–526.[9] L. Coutin, A. Lejay. Sensitivity of rough differential equations: an approach through the Omegalemma.
Preprint, (2017), HAL Id: hal-00875670.2410] H. Crauel, F. Flandoli,
Attractors for random dynamical systems.
Probab. Theory RelatedFields (1994), no. 3, 365–393.[11] A. M. Davie.
Differential equations driven by rough signals: an approach via discrete approxi-mation.
Appl. Math. Res. Express. AMRX , (2007), Art. ID abm009, 40.[12] L. H. Duc. Stability theory for Gaussian rough differential equations. Part II.
In preparation.[13] L. H. Duc, M. J. Garrido-Atienza, A. Neuenkirch, B. Schmalfuß.
Exponential stability ofstochastic evolution equations driven by small fractional Brownian motion with Hurst parameterin ( , Asymptotic stability for stochastic dissipative systems witha H¨older noise . Preprint. ArXiv: 1812.04556[15] P. Friz, M. Hairer.
A course on rough path with an introduction to regularity structure.
Uni-versitext, Vol.
XIV , Springer, Berlin, 2014.[16] P. Friz, N. Victoir.
Differential equations driven by Gaussian signals.
Ann. Inst. Henri.Poincar´e. Probab. Stat., Vol. (2), (2010), 369–413.[17] P. Friz, N. Victoir. Multidimensional stochastic processes as rough paths: theory and applica-tions.
Cambridge Studies in Advanced Mathematics, 120. Cambridge Unversity Press, Cam-bridge, 2010.[18] M. Garrido-Atienza, B. Maslowski, B. Schmalfuß.
Random attractors for stochastic equationsdriven by a fractional Brownian motion.
International Journal of Bifurcation and Chaos, Vol.20, No. 9 (2010) 27612782.[19] M. Garrido-Atienza, A. Neuenkirch, B. Schmalfuß.
Asymptotic stability of differential equationsdriven by H¨older-continuous paths
J. Dyn. Diff. Equat., (2018), in press.[20] M. Garrido-Atienza, B. Schmalfuss.
Ergodicity of the infinite dimensional fractional Brownianmotion.
J. Dyn. Diff. Equat., , (2011), 671–681. DOI 10.1007/s10884-011-9222-5.[21] M. Garrido-Atienza, B. Schmalfuss. Local Stability of Differential Equations Driven by H¨older-Continuous Paths with H¨older Index in ( , ) . SIAM J. Appl. Dyn. Syst. Vol. , No. , (2018),2352–2380.[22] M. Gubinelli. Controlling rough paths. J. Funtional Analysis , (1), (2004), 86–140.[23] M. Gubinelli, A. Lejay. Global existence for rough differential equations under linear growthconditions.
Prepirnt: hal-00384327, (2009), 20 pages.[24] M. Gubinelli, S. Tindel.
Rough evolution equations.
The Annals of Probability, Vol. , No.1, (2010), 1–75.[25] M. Hairer. Ergodicity of stochastic differential equations driven by fractional Brownian motion.
The Annals of Probability, Vol. , (2005), 703–758.[26] M. Hairer, A. Ohashi. Ergodic theory for sdes with extrinsic memory.
The Annals of Proba-bility, Vol. , (2007), 1950–1977.[27] M. Hairer, N. Pillai. Ergodicity of hypoelliptic sdes driven by fractional Brownian motion.
Ann.Inst. Henri Poincar´e, Vol. , (2011), 601–6282528] M. Hairer, N. Pillai. Regularity of laws and ergodicity of hypoelliptic stochastic differentialequations driven by rough paths.
The Annals of Probability, Vol. , (2013), 2544–2598.[29] Y. Hu. Analysis on Gaussian spaces.
World scientific Publishing, 2016.[30] Y. Hu, D. Nualart.
Rough path analysis via fractional calculus.
Transactions of the AmericanMathematical Society, Vol. , No. 5, (2009), 2689–2718.[31] R. Khasminskii.
Stochastic stability of differential equations.
Springer, Vol. 66, 2011.[32] A. Lejay.
Global solutions to rough differential equations with unbounded vector fields.
Preprint,HAL Id: irina-00451193.[33] T. Lyons.
Differential equations driven by rough signals.
Rev. Mat. Iberoam., Vol. (2),(1998), 215–310.[34] T. Lyons, M. Caruana, Th. L´evy. Differential equations driven by rpugh paths.
Lecture Notesin Mathematics, Vol. , Springer, Berlin 2007.[35] B. Mandelbrot, J. van Ness.
Fractional Brownian motion, fractional noises and applications.
SIAM Review, , No. 10, (1968), 422–437.[36] X. Mao, Stochastic differential equations and applications.
Elsevier, 2007.[37] D. Nualart, A. R˘a¸scanu.
Differential equations driven by fractional Brownian motion . Collect.Math. , No. 1, (2002), 55–81.[38] I. Nourdin. Selected aspects of fractional Brownian motion.
Bocconi University Press, Springer,2012.[39] S. Riedel, M. Scheutzow.
Roguh differential equations with unbounded drift terms.
J. DifferentialEquations, Vol. , (2017), 283–312.[40] L.C. Young.
An integration of H¨older type, connected with Stieltjes integration.
Acta Math. , (1936), 251–282.[41] M. Z¨ahle. Integration with respect to fractal functions and stochastic calculus. I.
Probab.Theory Related Fields. , No. 3, (1998), 333–374.[42] M. Z¨ahle.
Integration with respect to fractal functions and stochastic calculus. II.
Math. Nachr.225