| In this thesis,we mainly study the Schauder estimates for non-local Kolmogorov equations and applications to stochastic differential equations with jumps.Before giving the results,we introduce the following notations:For a Banach space B and T>0,we denote LT∞(B):=L∞([0,T];B),Lloc∞(B):=∩T>OLT∞(B),LT∞:=L∞([0,T]× Rd).Ⅰ.Schauder’s estimates for non-local Kolmogorov equationsIn terms of the Besov characterization of H(?)lder spaces,we develop a new method based on Littlewood-Paley’s decomposition and heat kernel estimates in integral form,to establish Schauder’s estimates for nonlocal Kolmogorov equations.Consider the usual heat equation,the key point of this method is the following integral form estimate of the heat kernel:for anyβ≥0 and some constant c=c(d,β)>0,(?) where Rj is the usual block operator in Littlewood-Paley’s decomposition,and ps(x)is the Gaussian heat kernel.(A)Schauder’s estimates for non-local equations with singular Levy measuresConsider the following non-local parabolic equations in Rd:(?) where b:R+×Rd→R is measurable,b·▽:=∑i=1d bi?i,and Lk,σ(α)is a nonlocal α-stable-like operator with form:(?)where z(α):=z1α∈(1,2)+z1|z|≤11α=1 with α∈(0,2),σ:R+×Rd→Rd?Rd is a measurable function,K(t,x,z)is symmetric in z and bounded from above and below,and v(α)is a nondegenerate α-stable Lévy measure which can be very singular(see Section 1.2).We make the following assumptions on κ,σ and b:(Hκβ)For some c0≥ 1 and β∈[0,1],it holds that for all t≥0 and x,y,z ∈ Rd,c0-1≤κ(t,x,z)≤c0,|κ(t,x,z)-κ(t,y,z)≤c0|x-y|β.and in the case of α=1,∫r≤|z|≤Rzκ(t,x,z)v(α)(dz)=0 for every 0<r<R<∞.(Hσγ)For some c0≥1 and γ∈[0,1],it holds that for all t≥0 and x,ξ∈Rd,c0-1|ξ|2≤|σ(t,x)ξ|2≤c0|ξ|2,‖σ(t,x)-σ(t,y)‖≤c0|x-y|γ.(Hbβ)For some c0≥1 and β∈[0,1],it holds that for all t≥0 and x,y ∈Rd with |x-y|≤1,|b(t,0)|≤c0,|b(t,x)-b(t,y)|≤c0|x-y|β.DEFINITION 1.We call a bounded continuous function u defined on R+×Rd a classical solution of PDE(6)if for some ε∈(0,1),u∈(∩M>0C(R+;l(α∨1)+ε(BM)))∩Lloc∞(l(α∨1)+ε(Rd)),and for all(t,x)∈[0,∞)× Rd,u(t,x)=∫0t(Lκ,σ)(α)u+b·▽u+f)(s,x)ds.Here,ls(Ω)denotes the H(?)lder space on the open set Ω with order s,BM denotes the ball with center zero and radius M.Under the assumptions above,we get the following result by using the heat kernel estimates in Littlewood-Paley form.THEOREM 1.Suppos that α∈(1/2,2),γ∈(1-α/α∨0,1],β∈(1-α)∨ 0,(α ∨ 1)γ),andα+β?N.Under(Hκβ),(Hσγ),and(Hbβ),for any f ∈Lloc∞(lβ),there is a unique classical solution u in the sense of Definition 1 of PDE(6)such that for any T>0 and some constant c=c(T,c0,d,α,β,γ)>0,‖u‖LT∞(lα+β)≤c‖f‖LT∞(lβ),‖u‖LT∞≤T‖f‖LT∞.(B)Schauder’s estimates for non-local kinetic equationsConsider the following degenerate nonlocal equation in R2d with Holder coefficients:(?) where u=u(t,x,v)and Lκ;ν(α)is a nonlocal α-stable-like operator with α∈(0,2)and kernel function κ,which acts on the variable v with the form:(?) where K(t,x,v,w)is symmetric in w and bounded from above and below,and b(t,x,v)takes the form b(t,x,v)=(b(1)(t,x,v),b(2)(t,x,v)):R+×Rd×Rd→Rd×Rd.We point out that since the multi-scale feature of kinetic equations,we consider the anisotropic Holder space and anisotropic Littlewood-Paley decomposition.For any s>0,let Cxs and Cvs be the global Holder-Zygmund spaces on x and v respectively,and Cas be the anisotropic Holder-Zygmund spaces(see Chapter 2).Define(?)We assume the following assumptions on κ and b:(Hβ,γ α,ν)For some c0≥1 and ν,β∈(0,1),it holds that for all t≥0 and x,v,w∈Rd,(?) and for some γ∈[β,1+α),(?) and for some closed and convex subset ??GLd(R),where GLd(R)is the set of all invertible d×d-matrices,▽vb(1)(t,x,v)∈?(8)DEFINITION 2.Let λ≥0.We call a bounded continuous function u defined on R+×R2d a classical solution of PDE(7)if for some ε∈(0,1),u∈C([0,∞);Cv(α∨1)+ε∩Cx1+ε),and for all t≥0 and x,v ∈ Rd,(?) Similarily,we haveTHEOREM 2.Let α ∈(1,2)and β∈(0,1),ν∈(0,β∧(α-1)),γ∈(1,1+α).Under(Hβ,γα,ν),for any f ∈Lloc∞(Cxγ/(1+α)∩Cvβ),there is a unique classical solution u in the sense of Definition 2 of PDE(7)such that for any T>0 and some c>0 being independent of A>0,(?)Ⅱ.Applications to stochastic differential equations with jumpsLet Lt(α)be a symmetric and rotationally invariant α-stable process with α∈(1,2)on some probability space(Ω,F,P).Consider the following degenerate SDE with jumps in R2d:dZs,t=b(t,Zs,t)dt+(0,σ(t,Zs,t)dLt(α)),Zs,s,=z∈R2d,t≥s≥0,(9)where σ:R+×R2d→Rd?Rd and b:R+×R2d→R2d are measurable functions satisfying(Hβ,γ α,ν)σ is Lipschitz continuous in x uniformly in t,and for some c0≥1 and all t≥0,(?) and for some ν,β∈(0,1)and γ∈(1,1+α),Moreover,(8)holds.First of all,under the assumptions above,we get the strong well-posedness of SDEs with multiplicative Levy noises.THEOREM 3.Let α∈(1,2),γ∈(1-α/2,1),ν∈(0,β∧(α-1)),and γ∈(1+α/2,1+α).Under(Hβ,γ α,ν),for each s≥0 and z ∈R2d,there exists a unique strong solution(Zs,t)t≥s to SDE(9).Next,we consider the C1-stochastic diffeomorphism flows property for SDE(9)with additive Levy noises.We introduce the following spaces:For a Frechet space F and time interval I,define C(I;F):={f:I→F is continuous},D(I;F):={f:I→F is càdlàg}.For k ∈ N0,let Ck be the Fréchet space of all k-order continuous differentiable functions with Fréchet metric:(?) We haveTHEOREM 4.Let α∈(1,2),γ∈(1+α/2,1+α)and β∈(1-α/2,1).Assume σ≡1 and b(t,x,v)=(v+b(1)(t,x),b(2)(t,x,v)),with b(1)∈ Lloc∞(Cxγ/(1+α)),b(2)∈Lloc∞(Cxγ/(1+α)∩Cvβ).Then the unique strong solution {Zs,t(z),t>s≥0,z ∈ R2d} of SDE(9)forms a C1-stochastic diffeomorphism flow.More precisely,there is a null set N such that for all ω?N,(ⅰ)For all 0≤s<r<t,it holds that Zs,t(z,ω)=Zr,t(Zs,r(z,ω),ω),?z ∈R2d,and z(?) Zs,t(z,ω)is a C1-diffeomorphism on R2d.(ⅱ)t (?)Zs,t(·,ω)∈ D([s,∞);C1)and s (?)Zs,t(·,ω)∈ D([0,t];C1).Then we apply this theorem to a random transport equation with Holder coefficients.THEOREM 5.Let α∈(1,2).Assume b ∈ C(R+×Rd)∩Lloc∞(Cγ)with γ∈(2+α/2(1+α),1).For any φ∈l1(Rd)and almost all ω,there is a unique function(t,x)(?)u(t,x,ω)∈C(R+;C0)∩ D(R+;C1)so that for each x ∈ Rd,t (?)u(t,x,ω)is absolutely continuous and?tu(t,x,ω)+(b(t,x)+Lt(ω))·▽xu(t,x,ω)=0,u(0,x)=φ(x). |
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