Transportation Cost Inequalities And Uniqueness Of Semigroups Generated By Diffusion Operators

This paper consists of two parts: one is about transportation inequalities, and another about uniqueness of semigroups generated by diffusion operators. Moreover, in the first part, two subparts are contained: Talagrand's T2-transportation inequality w.r.t. a uniform metric for diffusions; T2-transportation inequalities for infinite dimensional systems.We first introduce the concept of transportation inequality, which is related to the concentration of measure phenomenon.Let (E,d) be a metric space equipped with ο-field β such that d(·,·) is β × β-measurable. Given p ≥ 1, we say that the probability measure μ satisfies the L^p-transportation inequality on (E, d) if there exists a constant C > 0 such that for any probability measure v, the following inequality holdswhereis the Wasserstein metric of probability measures μ and v, and the infimum is taken over all probability measures Ï€ on the product space E × E with marginal distributions μ and v (saying coupling of (μ, ν)), and is the relative entropy of ν w.r.t. μ.To be short we write μ ∈ T_p(C) for this relation.In this paper, all the results on transportation inequalities are based on the paper from H. Djellout, A. Guillin, Liming. Wu about 《Transportation cost-information inequalities and applications to random dynamical systems and diffusions》 [3], in which the authors obtained T₂-transportation inequalities with respect to L²-metric for diffusions taking values from a finite dimensional space. However, it has a weak point, that is, the space C([0, N], R^d) is not complete with respect to L²-metric. It is well-known that C([0, N],R^d) is a Banach space with respect to a uniform metric. If we can obtain T2-transportation inequalities with respect to the uniform metric, it will be much better. It is out of this kind of consideration that the first chapter is born. Here we show the main result as follows:Consider the diffusion process = b(Xt)dt + c(Xt)dBt (19)where b : Rd —> Rd, and a : Rd -> Al^xn (the space of d x n matrices), and (Bt) is the standard Brownian Motion valued in Kn defined on some filtered probability space )- We assume that(Cl) \tr{a{x) - o(y)){a(x) - a(y)Y + (x - ytb(x) - 6(y)> < S\x - y\\ Vx,y € Rd, for some 8 > 0. Here A* denotes the transposition of the matrix A, tr(A) means the trace of the square matrix A and | ? | is the Euclidean metric on Kd.Consider the norm and the corresponding metric on C([0,N],]1/2sup -72||, V7l,72 € C([0,N),Rd).Theorem 2.3.1: Assume that b is locally Lipschitzian and a is globally Lipschitzian. If in addition, (Cl) holds and Halloo := sup{|<7(z)z|; x e M.d, \z\ < 1} < 00. Let Px be the law of the solution of (19) with the initial point Xq = x (Vz € Kd). Then there is some constant C > 0 such that Ps € T2(C) on C([0,iV],Rd) w.r.t. the uniform metric d above, for all initial point x € Rd and all N E N*.Notice that only those diffusions taking values from a finite dimensional space are considered in [3], what about the case with taking values from an infinite dimensional one? This is just what I want to discuss in the second chapter. Generalizing the results from a finite dimensional space to an infinite dimensional one, one has a common method, that is the way of approximation. We are of course without exception, and by this means develop the results for evolution stochastic differential equations. Later, we establish T%-transportation inequalities for a dissipative system by means of Yosida approximation, and give some applications to reaction-diffusion equations. The main results in the third chapter are as follows:Let H be a separable Hilbert space endowed with the norm | ? | induced by its inner product, U another separable Hilbert space.A Co semigroup is a strongly continuous one. The space of all Hilbert-Schmidt operators from U into H is denoted by L2(U,H), with the norm || ? U2. The operator norm is denoted by || ? ||.L2([0,T},H) = {/ : [0,T] -> H measurable ; \f\\ := /Qr \f(t)\2dt < +00}, TheL2-metric on L2([0,T],H) is defined as follows,,72) := l/^ |7lW " 72(0P^ , V71, 72 € L2([0,T],H) .Consider the Stochastic Differential Equation (SDE in abridge) in the Hilbert space HdX(t) = (AX(t) + F(X(t)))dt + B(X(t))dW(t), t > 0(20)where (Wt) is a cylindrical Brownian motion in a Hilbert space U with the covariance operator In;B{x) € L(U -?â–  #),Vz € H; F:B(F)(CH) ->ff; I ? I is the norm of H.For short, the Stochastic Differential Equation with coefficients A, F, B such as (20) is written as SDE(A,F,B).Hypothesis 2.1.(i) A is the infinitesimal generator of some Co semigroup (S(t))t>o on H; (ii) For any u E U, x -> B(x)u is continuous from H to H, and for any t > 0 and x E H, S(t)B(x) E L2(U —> #), and there is some nonnegative locally square-integrable function K(t) on K+ such that for all x,y E H,\\S(t)B(x)\\2# < M < 00.Hypothesis 2.2. (Dissipativity) There is some S > 0 such that for all x,y G(x - y, A(x -y) + F(x) - F(y)) + l-\\B(x) - B(y)\\\ < -S\x - y\\ (22)(L) B)(F) = H and F : H -? H is Lipschitzian, i.e., there exists some L > 0 such that\F(x)-F(y)\ 0, x e H,||e(n^n-nn5(nna:) - S(t)B(x)\\2 -> 0. (23)Then the SDE (20) has a unique L2-mild solution X(t) = X(t,x), i.e., X(t) is progressively measurable, supt 0 and satisfies for each t > 0 fixed,X(t) = S(t)x+ f S(t-s)[F(X(s))ds + B(X{s))dWs), F-a.s.JOMoreover this process X(t, x) possesses the following properties:(a) For all different initial points x,y € H,E\X(t,x)-X(t,y)\2 0. (24)In particular X(-) has a unique invariant probability measure fi such that for any initial measure v on H,W2{vPutx) < e-^z^), Vi > 0 (25)where Pt(x,â– ) := F(X(t,x) € â– ), W^^jA4) is the L2-Wasserstein distance between v and /x w.r.t. the | ? |-metric of H.(b) the probability distribution Fx (on L2([0,T],H)) of the mild solution X(-,x) starting from x of SDE(A, F, B)(20) satisfies the Talagrand transportation inequality T2(C) on L2([0,T],H) w.r.t. the L2-metric d2 for all x € # and T > 0, where the constant C is given byM2 Furthermore Pr(^; ?) € ^M—-) on (if, | ? |), as well as the unique invariant probabilitymeasure fx of (P<). (c) When U = H and B¹(x) exists and satisfiesM := sup US"1 (re) || = sup \B¹(x)z\ < +oo,H x,zï¿¡H;\z\=lthen the following Poincare inequality holdsM2M2 Var^f) := M/2) - /x(/)2 < -^^^(\BVf\% V/ G Clb(H). (26)The following results are given about the additive noise case, i.e. B(x) = B. Assume that K is a reflexive Banach space, densely and continuously embedded into H. We introduce the following hypotheses [15, p.80]. Hypothesis 2.4.(i) There exist 771, i^el such that the operators A — rji and F — 772 are m-dissipativeon H and 6 := —(rji + 772) > 0 (that is stronger than Hypothesis 2.2); (n) for some 771, 772 € 1R, the parts on K of A — 771 and jP — 772 are m-dissipative on K; (iii) B(F) D K and F maps bounded sets in K into bounded sets of H. Hypothesis 2.5. The process= f S(t-s)BdW(s), t>0, Joallows a continuous version in H, takes values in the domain 3(Fk) of the part of F in K, and for any T > 0,sup (\\WA(t)\\K + \\F(WA(t))\\K) <+oo, P-o.a.i€[0,T]Theorem 3.3.2: Assume that Hypotheses 2.1, 2.3, 2.4 and 2.5 are fulfilled. For every x ï¿¡ H, let Pz be the law of the unique generalized mild solution X(-,x) of the SDE (20) (a probability measure on C([0,T],H) C L2([0,Tj,#)), and Pt{x,dy) := F(X(t,x) 6 dy) the transition probability kernel. Then (a) For all x,y € H and t > 0,\X(tiX)-X(t,y)\ 0,(b) (T2-transportation inequality) Fx G T2(C) on C([0,T],#) w.r.t. the L2-metric d2 for all x G H and T > 0, where the constant C is given by (27)Moreover Py(a;,?) G T2 [ nt ) on H, as well as the unique invariant probability\ 25 )measure /j of(c) (log-Sobolev inequality) The law Fx satisfies the logS(C) with C = ||B||2/<52 on L2([0,T],tf), i.e., for all F € C}(L2([0,T\,H)),EF\X[0,T]) log j^^y < 2^E|VJP|l(X[0,r]), (28)where V is the gradient on the Hilbert space L2([0,T],H). Moreover, for every T > 0, Pt (/2log/2) - Pr/2logPr/2 < ^-Pr(|V/|2H), V/ 6 Cl(H) (29)on H, and p € logS(C) with C = \\B\\2/(26). (d) In particular when U = H and B^l exists and is bounded, then (30)As shown in [3, 66], many interesting consequences can be derived from the T2(C) of $x on L2([0,T],H) above. For instance :Corollary 3.3.3: Under the assumptions of Theorem 3.3.1 or Theorem 3.3.2, let C := M26², we have for any T > 0,(a) The following Poincare inequality holds for any F G Cl(L2([0,T],dt;H)),VarPx(F) := /0TJ?<** is the inner product in L2([0,T],ff), then|exp ^ sup [<7,fc) - ^l2/^) dPx < exp (^Ep^zj . (32)(c) (inequality of Hoeffding type) For any Lipschitz function V : H -t M such that its Lipschitzian coefficient ||V||x,iP < a, we have for all r > 0 and T > 0, (33)The third chapter has little connection with the previous two chapters. It is a completely independent system, in which the main tool we use is analysis technique, especially functional analysis. In this chapter, we discuss the uniqueness of semigroups generated by a diffusion operator, and give a sufficient and necessary condition of Ll{l, r)-uniqueness forthe Sturm-Liouville operator ï¿¡ = a(x)— + b(x)-----v(x). The main result is as follows:ax1 axGiven the following second order differential operatorCf = af" + bf - vflets(x) = e" ^ ^dtm(x) =a(x)s(x) wherea{x) > 0, v(x) > 0, x e (/,r)Theorem 4.3.1: Under the conditions above, the following two statements are equivalent:I). (C,Cg°{l,r)) is Pdl.r^mdx)- unique; II). There exists c G (/,r), such that,/ s{x) I m(y)(l + v(y))dydx = +oo, (34)and/ / (35)...