Stochastic Kinetic Equations And Ergodicity Of Supercritical Stochastic Differential Equations

The principal objectives of this dissertation are two parts:(1)We consider the following stochastic kinetic equation(SKE)of It(?)’s type where and the unknown u is a function of(t,ω,x,v)with that(x,v)represents the position and velocity,the nonhomogenous or free terms(f,g):R+× Ω × R2d→(R,l2)are P×B(R2d)-measurable processes.Notice that SKE(*)are highly degenerate in x-direction.We establish the optimal regularity estimates for the Cauchy problem of SKE(*)with random coefficients in anisotropic Besov spaces under some H(?)lder regularity assumptions on a,b,σ.Our optimal regularity estimates are not only for nondegenerate velocity component v,but also for degenerate position component x.When a is nonrandom,by Duhamel’s formulation,and using certain properties of the heat kernel of operator Δv+v·▽x,one can establish a satisfactory Schauder theory([37]).Next,by a generalized It(?)-Wentzell’s formula(see[44,45]),we transform SKE(*)with random but constant a,σ into a model equation,i.e.,a is a constant scalar matrix and then obtain the regularity estimates for(*),where the key point is that we need to work in Besov spaces with finite integrability index.Finally,we use the freezing coefficient argument to derive the optimal regularity estimates for variable random coefficients a,σ.In order to make the perturbation term can be absorbed by the freezing term,we introduce a new localized anisotropic Besov norm.As applications,we study the following two problems.Concretely,(ⅰ)We study the nonlinear filtering problem for a degenerate diffusion process,and obtain the existence of the conditional probability densities under a few assumptions.Moreover,we prove the conditional probability densities satisfy certain SKEs and then obtain the regularity of conditional probability densities as above.(ⅱ)We show the well-posedness for a class of the following super-linear growth stochastic kinetic equations driven by velocity-time white noises where B(t,v)is the velocity-time white noise,(x,v)∈ R2.In particular,it includes a kinetic version of continuous Parabolic Anderson Model when γ=0 and u≥0 a.s.Furthermore,we also present some regularity estimates on t,x,v.(2)We consider the stochastic differential equation(SDE)driven by an α-stable process dXt=b(Xt)dt+σ(Xt-)dLtα,X0=x ∈Rd,where b:Rd→Rd and σ:Rd→Rd(?)Rd are locally γ-H(?)lder continuous withγ∈((1-α)+,1],and Ltα is a d-dimensional symmetric rotationally invariant α-stable process with α∈(0,2).Under certain dissipative and non-degenerate assumptions on b and σ,we show the V-uniformly exponential ergodicity for the semigroup Pt associated with {Xt(x),t≥0}.Our proofs are mainly based on the heat kernel estimates recently established in[58]to demonstrate the strong Feller property and irreducibility of Pt.Interestingly,when α tends to zero,the diffusion coefficient σ can increase faster than the drift b.As an application,we put forward a new heavy-tailed sampling scheme.In detail,let μ(dx)=e-U(x)dx/∫Rd e-U(x)dx,where U:Rd→R is a continuous function.We suppose that there are β,C0>0 such that for all |x|≥ 1,U(x)≥(d+β)log |x|-C0.The above assumption means that e-U(x)has a polynomial decay rate as |x|→∞,which characterizes a heavy-tailed distribution,as opposed to the exponential or light-tailed one.We consider the following SDE where(?)α and B are independent of U(see Section 4.3).We prove SDE(**)is ergodic and the law of the solution Xt exponentially converges to μ in some sense as t→∞.Thus,one can sample μ theoretically from Xt when t is large that is one can construct a sampling algorithm by the Euler discretization.