Stationary Solutions Of Stochastic Partial Differential Equations And Infinite Horizon Backward Doubly Stochastic Differential Equations

In this thesis we study the existence of stationary solutions for stochastic partial differential equations.We establish a new connection between solutions of backward doubly stochastic differential equations(BDSDEs)on infinite horizon and the station-ary solutions of the SPDEs.For this,we prove the existence and uniqueness of the L_ρ²(R^d;R¹)(?)L_ρ²(R^d;R^d)valued solutions of BDSDEs with Lipschitz nonlinear term on both finite and infinite horizons,so obtain the solutions of initial value problems and the stationary weak solutions(independent of any initial value)of SPDEs.Also the L_ρ²(R^d;R¹)(?)L_ρ²(R^d;R^d)valued BDSDE with non-Lipschitz term is considered.More-over,we verify the time and space continuity of solutions of real-valued BDSDEs,so obtain the stationary stochastic viscosity solutions of real-valued SPDEs.The connec-tion of the weak solutions of SPDEs and BDSDEs has independent interests in the areas of both SPDEs and BSDEs.The thesis is organized as follows:Chapter 1 Introduction gives an introduction for the stationary solutions of random dynamical systems.Let u:[0,∞)×U×Ω→U be a measurable random dynamical system On a measurable space(U,B)over a metric dynamical system(Ω,F, P,(θ_t)_t≥0),then a stationary solution is a F measurable random variable Y:Ω→U such that(Arnold[1])u(t,Y(ω),ω)=Y(θ_tω)for all t≥0 a.s.. Then we give an simple nontrivial example,Ornstein-Uhlenbeck process.We indicate a pathwise stationary solution describes the pathwise invariance of the stationary so- lution over time along the measurable and P-preserving transformationθ_t:Ω→Ωand the pathwise limit of the solutions of random dynamical systems.In contrast to the deterministic dynamical systems,also due to the fact that the external random force exists at all time,the existence of stationary solutions of stochastic dynamical systems generated e.g.by stochastic differential equations(SDEs)or stochastic par-tial differential equations(SPDEs),is a difficult and subtle problem.We point out in Chapter 1 that there have been extensive works on stability and invariant manifolds of random dynamical systems,and researchers usually assume there is an invariant set (or a single point:a stationary solution or a fixed point,often assumed to be 0),then prove invariant manifolds and stability results at a point of the invariant set(Arnold [1]and references therein,Ruelle[48],Duan,Lu and Schaumulfuss[18],[19],Li and Lu [31],Mohammed,Zhang and Zhao[38]to name but a few).But the invariant mani-folds theory gives neither the existence results of the invariant set and the stationary solution nor a way to find them.In particular,for the existence of stationary solutions for SPDEs,results are only known in very few cases([13],[20],[38],[50],[51]).In[50], [51],the stationary strong solution of the stochastic Burgers' equations with periodic or random forcing(C³ in the space variable)was established by Sinai using the Hopf-Cole transformation.In[38],the stationary solution of the stochastic evolution equations was identified as a solution of the corresponding integral equation up to time +∞and the existence was established for certain SPDEs by Mohammed,Zhang and Zhao.But the existence of solutions of such a stochastic integral equation in general is far from clear.The main purpose of this thesis is to find the stationary solution of the following SPDE Here B is a two-sided cylindrical Brownian motion on a separable Hilbert space U₀.The above SPDE is very general,especially when we consider the weak solution of it,the nonlinear functions f and g can includeâ–½v and the second order differential operator L is allowed to be degenerate,while in most literature,g is not allowed to depend onâ–½v or g only depends onâ–½v linearly(Da Prato and Zabczyk[16],Cyongy and Rovira [23],Krylov[27],Mikulevicius and Rozovskii[37],Pardoux[41]).As an intermediate step,the result of existence and uniqueness of the weak solutions,obtained by solving the corresponding backward doubly stochastic differential equations(BDSDEs)under weak Lipschitz condition,appears also new.In Chapter 1,we also explain the reason why the stationary solution we study in this thesis gives the support of the corresponding invariant measure,so reveals more detailed information than an invariant measure.BDSDEs will be used as our tool to study stationary solutions of SPDEs and we will prove that the solutions of the corresponding infinite horizon BDSDEs give the desired stationary solutions of SPDEs,so in Chapter 1 we briefly recall the development of BSDE and BDSDE since Pardoux and Peng's pioneering work[42]in 1990.As far as we know,the connection of the pathwise stationary solutions of the SPDEs and infinite horizon BDSDEs we study in this thesis is new.Chapter 2 The Correspondence Between Stationary Solutions of SPDEs and BDSDEs shows us how to obtain stationary weak solutions of SPDEs through its corresponding BDSDEs.For this,we first apply a "perfection procedure" to the solution of general BDSDE. Theorem 2.2.4.Assume in the Hilbert space H,the following BDSDE has a unique weak solution(Y,Z),then under Conditions(A.2.1)and(A.2.2),(Y_t,Z_t)_t≥0 is a "perfect" stationary solution,i.e. The generalized equivalence of norm principle which is a simple extension of the equiv-alence of norm principle obtained by Kunita([28]),Barles and Lesigne([4]),Bally and Matoussi([3])to random functions,plays an important role in the analysis through the thesis. Lemma 2.3.3.(generalized equivalence of norm principle)If s∈[t,T],φ:Ω×R^d→R¹ is independent of F_t,s^W andφρ^-1∈L¹(Ω(?)R^d),then there exist two constants c>0 and C>0 such that Moreover ifψ:Ω×[t,T]×R^d→R¹,ψ(s,·)is independent of F_t,s^W andψ_p^-1∈L¹(Ω(?)[t,T](?)R^d),then Then we take weak solution as an example and transfer the stationary property from BDSDE to the corresponding SPDE by "perfection procedure" and "generalized equiv-alence of norm principle". Theorem 2.3.13.Under Conditions(A.2.1)'-(A.2.4)',for arbitrary T and t∈[0,T], let v(t,·)(?)Y_T-t^T-t,where(Y_·^t,·,Z_·^t,)is the L_ρ²(R^d;R¹)(?)L_ρ²(R^d;R^d]valued solution of BDSDE with(?)_s=B_T-s-B_T for all s≥0.Then v(t,·)is a "perfect" stationary weak solution of the following SPDEChapter 3 Stationary Weak Solutions of SPDEs aims to Study L_ρ²(R^d;R¹)(?) L_ρ²(R^d;R^d)valued BDSDEs and the stationary weak solutions of its corresponding SPDEs.One of the necessary intermediate steps is to study the BDSDEs on finite horizon and establish the connection between their solutions and the weak solutions of SPDEs.Our method to study the L_ρ²(R^d;R¹)(?)L_ρ²(R^d;R^d)valued solutions of BDS-DEs on finite horizon was inspired by Bally and Matoussi's approach on the exis-tence and uniqueness of solutions of BDSDEs with finite dimensional Brownian mo-tions([3]).But our results are stronger and our conditions are weaker.We will solve the BDSDEs driven by the cylindrical Brownian motion and nonlinear terms satisfying Lipschitz conditions in the space L_ρ²(R^d;R¹).We obtain a unique solu-tion.The result Y_·^t,·∈S^2,0([t,T];L_ρ²(R^d;R¹)),which plays an important role in solving the nonlinear BDSDEs and proving the connection with the weak solutions of SPDEs(also BSDEs and PDEs), was not obtained in[3].We believe our results for finite horizon BDSDEs are new even for BSDEs.The main result for the finite horizon BDSDEs with infinite dimensional noise is Theorem 3.1.2.Under Conditions(H.3.1)-(H.3.4),the following L_ρ²(R^d;R¹)(?)L_ρ²(R^d;R^d) valued BDSDE. has a unique solution. The relationship between this BDSDE and its corresponding finite horizon SPDE with infinite dimensional noise is established in the following theorem. Theorem 3.2.3.Under Conditions(H.a.1)-(H.3.4),if we define u(t,x)=Y_t^t,x,where (Y_s^t,x,Z_s^t,x)is the solution of BDSDE in Theorem 3.1.2,then u(t,x)is the unique weak solution of the following SPDE Moreover, With the results of finite horizon BDSDEs,we prove the existence and uniqueness of L_ρ²(R^d;R¹)(?)L_ρ²(R^d;R^d)valued solution of infinite horizon BDSDEs. Theorem 3.3.1.Under Conditions(H.3.4)-(H.3.7),the following L_ρ²(R^d;R¹)(?)L_ρ²(R^d;R^d) valued BDSDE has a unique solution. We have given the stationary weak solutions of the corresponding SPDEs in Theorem 2.3.13,but in the proof of Theorem 2.3.13 we used the following two theorems without proofs in order to obtain the continuity of solution of SPDE.We prove them in this chapter. Theorem 2.3.10.Under Conditions(A.2.1)'-(A.2.4)',the BDSDE in Theorem 2.3.13 has a unique Solution(Y_s^t,x,Z_s^t,x.Moreover. Theorem 2.3.11.Under Conditions(A.2.1)'-(A.2.4)',let u(t,·)(?)Y_t^t,·,where(Y_·^t,·,Z_·^t,·) is the solution of the BDSDE in Theorem 2.3.13.Then for arbitrary T and t∈[0,T], u(t,·)is a weak solution for the following SPDE Moreover,u(t,·)is a.s.continuous w.r.t,t in L_ρ²(R^d;R¹).Chapter 4 Non-Lipschitz Condition further discusses L_ρ²(R^d;R¹)(?)L_ρ²(R^d;R^d) valued BDSDE and its corresponding SPDE with linear growth non-Lipschitz nonlinear term.Besides the monotone condition,we make a use of Lepeltier and San Martin's work to deduce the following proposition,which plays an important role in proving the results for finite horizon BDSDEs with non-Lipschitz term. Proposition 4.2.4.Given(U.(·),V.(·))∈S^2,0([0,T];L_ρ²(R^d;R¹))(?)M^2,0([0,T];L_ρ²(R^d;R^d)), then under Conditions(H.4.1)-(H.4.7),the following L_ρ²(R^d;R¹)(?)L_ρ²(R^d;R^d)valued BDSDE has a unique solution. Then following a similar procedure in Chapter 3,we establish the connection between L_ρ²(R^d;R¹)(?)L_ρ²(R^d;R^d)valued solutions of finite horizon BDSDEs and weak solutions of SPDEs.After that,we solve the infinite horizon BDSDEs and study the continuity of the solution.Although in this chapter we use the non-Lipschitz condition to weaken the Lipschitz continuous condition used in Chapter 3,the procedure is rather similar to Chapter 3.So we don't intend to list all the results,and only give the eventual theorem in which we have the stationary weak solution of SPDE under non-Lipschitz condition. Theorem 4.1.4.Under Conditions(A.4.1)'-(A.4.6)',for arbitrary T and t∈[0,T], Iet v(t,·)(?)Y_T-t^T-t,·,where(Y_·^t,·,Z_·^t,·)is the solution of the BDSDE in Theorem 2.3.13. Then v(t,·)is a "perfect" stationary weak solution of the SPDE in Theorem 2.3.13.Chapter 5 Stationary Stochastic Viscosity Solutions of SPDEs shows us how to obtain the stationary stochastic viscosity solutions of SPDEs through the cor-respondence between the real-valued BDSDEs and the stochastic viscosity solutions of SPDEs.We first recall Buckdahn and Ma's idea to define the stochastic viscosity solutions of SPDEs by the Doss-Sussmann transformation,then we prove the existence and uniqueness of solutions of general real-valued BDSDEs on infinite horizon. Theorem 5.2.4.Under Conditions(H.5.1)-(H.5.3),the following BDSDE has a unique solutionComparing the stochastic viscosity solution with the weak solution,we need more information for the stochastic viscosity solution.In particular,the space continuity of solution of BDSDE as well as time continuity is considered. Proposition 5.3.2.Under Conditions(A.5.1)-(A.5.4),let(Y_s^t,x_s≥0be the solution of the following BDSDE then for arbitrary T and t∈[0,T],x∈R^d,(t,x)→Y_t^t,xis a.s.continuous. The space continuity and time continuity of solution are transferred from BDSDE to SPDE after the connection between the real-valued BDSDEs and the stochastic viscosity solutions of SPDEs is established. Theorem 5.3.3.Under Conditions(A.5.1)-(A.5.4),for arbitrary T and t∈[0,T], x∈R^d,let v(t,x)(?)Y_T-t^T-t,x,where(Y_s^t,x,Z_s^t,x)is the solution of BDSDE in Proposition 5.3.2.Then v(t,x)is continuous w.r.t,t and x and is a stochastic viscosity solution of the following SPDE With "perfection procedure",we have the stationary stochastic viscosity solutions of SPDEs. Theorem 5.3.4.Under Conditions(A.5.1)-(A.5.4),for arbitrary T and t∈[0,T], let v(t,x)(?)Y_T-t^T-t,x,where(Y_s^t,x,Z_s^t,x)is the solution of BDSDE in Proposition 5.3.2. Then v(t,x)is a "perfect" stationary stochastic viscosity solution of SPDE in Theorem 5.3.3. Finally,we would like to point out that although the techniques in Chapter 4 can be similarly applied to studying the stochastic viscosity solutions of SPDEs with linear growth non-Lipschitz term,we don't intend to include the analysis and only deal with the Lipschitz condition in this chapter.