Ξ44fflΞ5» ο ffi fi $ Vol.44, No.5 2015 9" ADVANCES IN MATHEMATICS(CHINA) Sep., 2015 doi: 10.11845/sxjz.2014024b The Continuity of SDE With Respect to Initial Value in the Total Variation PENG Xuhui (1. College of Mathematics and Computer Science, Hunan Normal University, Changsha, Hunan, 410081, P. R. China; 2. Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, P. R. China;) Abstract: Let X t(x) be the solution of stochastic differential equation driven by Brownian motion, here x is the initial value. If the Hörmander s condition holds and the solution globally exists, we prove that the law of X t(x) is continuous in variable x with respect to the total variation distance. Keywords: stochastic differential equations; Hörmander s condition; strong Feller property MR(2010) Subject Classification: 60H10; 60H07 / CLC number: O211.63 Document code: A Article ID: 1000-0917(2015)05-0783-06 0 Preinaries Let W be the space of all continuous functions from R + := [0, )tor m vanishing at starting point 0, which is endowed with the locally uniform convergence topology and the Wiener measure μ W so that the coordinate process W t (ω) =ω t is a standard m-dimensional Brownian motion. Let H W be the Cameron-Martin space consisting of all absolutely continuous functions with square integrable derivatives. The inner product in H is denoted by h 1,h 2 H := m i=1 0 ḣ i 1(s)ḣi 2(s)ds. The triple (W, H,μ W ) is also called the classical Wiener space. Let D be the Malliavin derivative operator. For k N and p 1, let D k,p be the associated Wiener-Sobolev space with the norm F k,p := F p + DF p + + D k F p, where p is the usual L p -norm. Let X : W R d be a smooth Wiener functional in k,p Dk,p and Σ X ij := DXi,DX j H be the Malliavin covariance matrix of X. Received date: 2014-02-26. Foundation item: This work was supported by the Youth Scientific Research Fund of Hunan Normal University (No. Math140650) and the Scientific Research Foundation for Ph.D Hunan Normal University (No. Math140675). E-mail: pengxuhui@amss.ac.cn
784 ν fl ff # 44ffl 1MainResults Let X n and X be d-dimensional random variables on (W, H,μ W ). In [2, Corollary 9.6.12] Bogachev proved if X n X in D 1,p (p d) and for almost all ω, {D h X(ω), h H} = R d, then the laws of X n converge to the law of X in total variation distance. In [5, Theorem 1.1], Dong et al. proved that if X n converge to X in probability, X n,x D 2,p (p>1), and sup X n 2,p <, Σ X is almost invertible, n 0 then the laws of X n converge to the law of X in total variation distance. In [5, Theorem 1.1], p>1 is needed. But in [2, Corollary 9.6.12], p>dis needed. There are also many other papers concerned about the convergence of random variables, for example [1][6] etc. In this article, we consider the following SDE: dx t = b(x t )dt + A(X t )dw t, X 0 = x, (1.1) where b : R d R d and A : R d R d m are locally Lipschitz and smooth. Consider the following vectors fields on R d associated with (b, A), A(x) =(a ij (x)), i =1, 2,,d, j =1, 2,,m, d A j = a ij (x), j =1, 2,,m, x i i=1 A 0 = b 1 2 m A l A l, l=1 here A l A l = d i,j=1 a il(x) xi a jl (x) x j. We say that (b, A) satisfies Hörmander s condition at point x R d if the vector space spanned by the vector fields {A j } j=1,2, m, {[A i,a j ] m i,j=0 }, {[[A i,a j ],A k ]} m i,j,k=0, at point x is R d.here[a i,a j ] denotes the Lie bracket between A i and A j. The generator of X t is given by where L = i b i (x) x i + 1 2 2 σ ij (x), x i x j i,j σ ij =(A A ) ij. Let H : R d R + be a C -function with x H(x) =, which is called a Lyapunov function. We assume that for some Lyapunov function H, LH ch. (1.2)
5» fiφ: The Continuity of SDE With Respect to Initial Value in the Total Variation 785 Our main work is to prove the following theorem by using [5, Theorem 1.1]. Theorem 1.1 If for any point x R d,(b, A) satisfies Hörmander s condition at point x and the solution to (1.1) globally exists, then the law of X t (x) is continuous in variable x with respect to the total variation distance. In particular, the semigroup (P t ) t>0 has the strong Feller property, i.e., for any t>0andf B b (R d ), x Ef(X t (x)) is continuous. Remark 1.1 If b, A C and (b, A) satisfies Hörmander s condition at any point x R d, then it is well known that P t is strong Feller. For example, see [7]. For the stochastic differential equations driven by degenerate subordinated Brownian motions, Zhang [9 10] made some researches on the existence and smoothness of the density of solutions. By [7, Theorem 5.9], if (1.2) holds for some Lyapunov function H and c>0, then the solution to (1.1) globally exists, so we have the following corollary. Corollary 1.1 If (b, A) satisfyhörmander s condition at any point x R d,andforsome Lyapunov function H and c>0, (1.2) holds, then the law of X t (x) is continuous in variable x with respect to the total variation distance. The article is organized as follows. In Section 2, we give the proof of Theorem 1.1, and in Section 3, we give an example. 2 Proof of Theorem 1.1 Before we give the proof of Theorem 1.1, we need some notations and a lemma. Let B M := {x R d : x <M}, B M := {x R d : x M},BM c = {x Rd : x M} for any M>0. For any n N, letχ n (x) be a cut-off function on [0, ) with χ n Bn =1, χ n B c n+1 =0, 0 χ n 1, and set We have b n (x) =b(x)χ n (x), A n (x) =A(x)χ n (x). b n,a n C b (R d ). Consider the following SDE: dx n t (x) =b n(x n t (x))dt + A n(x n t (x))dw t, X n 0 = x. (2.1) Define τ n (x) :=inf { t 0: X t (x) n }. By the uniqueness of the solution to SDE, for any x R d, n N, X n t (x) =X t (x), t<τ n (x) a.s. (2.2)
786 ν fl ff # 44ffl X n+1 t (x) =X n t (x), t<τ n(x) a.s. (2.3) Now we introduce a lemma which will be used in the proof of Theorem 1.1. The proof of this lemma is the same to [4, Lemma 4.1]. But for the convenience of reading, we prove it again. Lemma 2.1 For any n N and x B n, sup I {t τn(y)} I {t τn 1(x)} a.s. Proof Let Γ be a measurable set with P(Γ c )=0andsuchthatXs n (x, ω) is continuous with respect to s and x for ω Γ. For ω Γ, the conclusion is apparent if sup I {t τn(y)}(ω) =0 or I {t τn 1(x)}(ω) =1. Assume that sup I {t τn(y)}(ω) =1andI {t τn 1(x)}(ω) =0,then sup Xs n 1 (x, ω) n 1. (2.4) Furthermore, by (2.3), sup Xs n (x, ω) n 1. (2.5) Since sup I {t τn(y)}(ω) = 1, then there exist a sequence {y k } R d with y k x as k,andfork large enough, sup Xs n (y k,ω) n. (2.6) Since [0,t] B n [0, ) R d is a compact set, Xs n (x, ω) is an uniformly continuous function on [0,t] B n. Hence, for ε 0 = 1 2,thereexistsδ 0 > 0 such that for any y x δ 0 and any s [0,t], Xs n (y, ω) Xs n (x, ω) ε 0 = 1 2. That means when y x δ 0, sup Xs n (y, ω) sup Xs n (x, ω) + 1 2, which contradicts (2.5) and (2.6). Now we are in a position to give the proof of Theorem 1.1. ProofofTheorem1.1 Let f be a bounded measurable function. For any x, y B n, E[f(X t (x)) f(x t (y))] E[f(X t (x))i t<τn(x) f(x t (y))i t<τn(y)] + f P(t τ n (x)) + f P(t τ n (y)) E[f(X n t (x))i t<τn(x) f(x n t (y))i t<τn(y)] + f P(t τ n (x)) + f P(t τ n (y)) E[f(X n t (x)) f(x n t (y))] +2 f P(t τ n (x)) + 2 f P(t τ n (y)).
5» fiφ: The Continuity of SDE With Respect to Initial Value in the Total Variation 787 Since b n = b, A n = A on B n,(b n,a n ) also satisfy Hörmander s condition at points x and y. So the Malliavin matrices of Xt n(x),xn t (y) are almost invertible. Since b n,a n Cb (Rd ), then for any p>0, sup DXt n (u) 1,p <, u B n By [5, Theorem 1.1] or [2, Corollary 9.6.12], So, by (2.7) and Lemma 2.1, sup f 1 sup f 1 sup DXt n (u) 2,p <. u B n E[f(Xt n (x)) f(xn t (y))] =0. (2.7) E[f(X t (x)) f(x t (y))] Letting n in the above inequality, we obtain 3Example 2P(t τ n (x)) + 2 sup P(t τ n (y)) 2P(t τ n (x)) + 2P(t τ n 1 (x)). sup f 1 E[f(X t (x)) f(x t (y))] =0. In the end of this article, we give a example. The following stochastic oscillators are studied in [3, 8] etc. dz i (t) =u i (t)dt, i =1, 2,,d, du i (t) = zi H(z(t),u(t))dt, i =2, 3,,d 1, du i (t) = [ zi H(z(t),u(t)) + γ i u i (t)]dt + (3.1) T i dwt i, i =1,d, where d N, γ 1,γ d R, T 1,T d > 0, and H(z,u):= d i=1 ( ) 1 2 u i 2 + V (z i ) U(z i+1 z i ). d 1 + As that in [8], we assume (H1) Growth at infinity. U and V are C and there exist real constants l 1, k max{2,l}, a k,b l > 0 such that λ λ k U(λx) =a k x k, λ λ1 k U (λx) =ka k x k 1 sign(x), λ λ l V (λx) =a k x l, λ λ1 l V (λx) =ka l x l 1 sign(x), (H2) Non-degeneracy. For any q R, thereexistsn = n(q) 2 such that n U(q) 0. Proposition 3.1 Assume that V,U C (R) are nonnegative functions such that (H1) and (H2) hold, then the law of (z t,u t ) are continuous with the initial value in the total variation distance. i=1
788 ν fl ff # 44ffl Proof It can be directly obtained by Theorem 1.1 and [8, Theorem 1.1 and Section 3]. Acknowledgements The author is very grateful to Dong Zhao, Song Yulin and Zhang Xicheng for their quite useful conversations. References [1] Bally, V. and Caramellino, L., On the distance between probability density functions, Electron. J. Probab., 2014, 19: Article 110, 33 pages. [2] Bogachev, V.I., Differentiable Measures and the Malliavin Calculus, Math. Surveys Monogr., Vol. 164, Providence, RI: AMS, 2010. [3] Carmona, P., Existence and uniqueness of an invariant measure for a chain of oscillators in contact with two heat baths, Stochastic Process. Appl., 2007, 117(8): 1076-1092. [4] Dong, Z. and Peng, X.H., Malliavin matrix of degenerate SDE and gradient estimate, Electron. J. Probab., 2014, 19: Article 73, 26 pages. [5] Dong, Z., Peng, X.H., Song, Y.L. and Zhang, X.C., Strong Feller properties for degenerate SDEs with jumps, 2013, arxiv: 1312.7380. [6] Malicet, D. and Poly, G., Properties of convergence in Dirichlet structures, J. Funct. Anal., 2013, 264(9): 2077-2096. [7] Rey-Bellet, L., Ergodic properties of Markov processes, In: Open Quantum Systems II, Lecture Notes in Math., Vol. 1881, Berlin: Springer-Verlag, 2006, 1-39. [8] Rey-Bellet, L. and Thomas, L.E., Asymptotic behavior of thermal nonequilibrium steady states for a driven chain of anharmonic oscillators, Comm. Math. Phys., 2010, 215(1): 1-24. [9] Zhang, X.C., Densities for SDEs driven by degenerate α-stable processes, Ann. Probab., 2014, 42(5): 1885-1910. [10] Zhang, X.C., Derivative formulas and gradient estimates for SDEs driven by α-stable processes, Stochastic Process. Appl., 2013, 123(4): 1213-1228. 829/.)4,/'1>*A?6&(-7:,5<; CDB (1. μπλflνflψωßψ flfl!,,, 410081; 2. %± fl!νflψffρ flχffi!, fl, 100190) @= U X t (x) hmgyonanmetgqpjmw, x h^kr. cs Hörmander f VIZ`PJMWbXLp, Eiq] X t (x) MQGRnKr x pbfhodj[l. 03+ etgqpj; Hörmander fv; _ Feller k