Asymptotic log-Harnack inequality and applications for SPDE with degenerate multiplicative noise
Let \((H, (.,.), \big\| .\|_{H})\) be a real separable Hilbert space and suppose that \(A\) is a non-negative self-adjoint linear operator and \(V = D\big(A^{\frac12}\big)\) is a reflexive Banach space with the norm \(\big\|u\big\|_{V} = \big\|A^{\frac12} u\big\|_{H}\), which is continuously and densely embedded into \(H\). Assume that there exists an orthonormal basis \(\{ e_k \}_{k \geq 1}\) for \(H\) of eigenfunctions of \(A\), and the associated increasing eigenvalue sequence \(0 <\lambda_1\leq \lambda_2 \leq \cdots \leq \lambda_n \leq \cdots \uparrow \infty\). We thus obtain a Gelfand triple \[V \subset H \equiv H^* \subset V^{*}, \] where \(V^{*}\) (respectively \(H^*\)) is the dual space of \(V\) (resp. \(H\)). The isomorphism \(\equiv\) between \(H\) and \(H^*\) follows from the Riesz representation theorem. Let \(\langle .,. \rangle\) denote the dualization between \(V\) and \(V^{*}\) , it follows that \[\langle u,v \rangle=(u,v),\ u \in H,\ v \in V.\] Let \(\Big(\Omega, \mathcal{F},\big(\mathcal{F}_t\big)_{t \geq 0}, \mathbb{P}\Big)\) be a complete filtered probability space satisfying the usual condition and \((L_2(H , H) , \big\|.\big\|_{L_2})\) be the space consisting of all Hilbert-Schmidt operators from \(H\) to \(H\). For any \(u \in H\), we denote \(u_k := (u , e_k), k \geq 1\). For any \(N \in N\) , we define a projection \(P_N:H \longrightarrow H\) by \[P_Nu :=\sum_{|k| \leq N}u_k e_k ,\quad u \in H .\] The authors consider the following stochastic evolution equation: \[du(t)=[-Au(t)+F(u(t))]dt+B(u(t))dW(t),\quad u(0)=x \in H,\tag{1.1}\] where \(W(t)\) is a cylindrical Wiener process in \(H\) defined on \(\Big(\Omega, \mathcal{F},\big(\mathcal{F}_t\big)_{t \geq 0},\mathbb{P} \Big)\). The function \(F : V\longrightarrow V^*\) and \(B: V\longrightarrow L_2(H,H)\) are measurable and satisfy the following conditions. (1) For all \(u,v,w \in V\), \(s\longmapsto \langle F(u+sv),w \rangle\) is is continuous on \(\mathbb{R}\). (2) For all \(u,v\in V,\) There exist constants \(K_1,C\) and \(\gamma\) such that \[2\langle F(u)-F(v),u-v \rangle \leq K_1\big\|u-v\big\|_{H}^2,\] and \[\big\|F(u)\big\|_{V^*} \leq C(1+\big\|u\big\|_{V}) (1+\big\|u\big\|_{H}^{\gamma}).\] (3) \(B\) is bounded and Lipschitz. The Lipschitz constant is \(\leq K_2\). (4) For all \(u \in V,\) there exists a constant \(N_0 \in \mathbb{N}\) such that for all \(u \in H\), we have \(P_{N_0}H \subset \textrm{Range}(B(u))\) and \(B(u)x=0\) if \(x \in (I-P_{N_0})H.\) Moreover, the corresponding pseudo-inverse operator \(B(u)^{-1} : P_{N_0} H \longrightarrow P_{N_0} H\) is uniformly bounded. From (2)--(3), there exist constants \(\beta_1\) and \(\beta_2\) such that following coercivity condition holds \[ 2 \langle -Au+F(u),u \rangle +\big\|B(u)\big\|_{L_2}^2 \leq \beta_1 + \beta_2 \big\|u\big\|_{H}^2-2 \big\|u\big\|_{V}^2.\tag{2.2}\] We recall that a continuous \(H\)-valued \(( \mathcal{F}_t)\)-adapted process \(\{ u(t) \}_{t \in [0,T]}\) is called a strong solution of (1.1) if for its \(dt \otimes P\)-equivalence class we have \(u \in L^2([0,T]\times \Omega, dt \otimes \mathbb{P}; V)\), and \[u(t)=x+\int_{0}^{t}\big(-Au(s)+F(u(s))\big) ds +\int_{0}^{t}B(u(s)) dW(s), \ t \in [0,T]\] holds \(\mathbb{P}\)-.a.s. \noindent Under the previous assumption, there exists a unique solution of \((1.1)\), we denote it by \(u^{x}(t)\) and its transition semigroup is given by \[P_tf(x)=\mathbb{E}(u^{x}(t)), \quad f \in \mathcal{B}_b(H),\] \(\mathcal{B}_b(H)\) stand for the space of bounded measurable functions on \(H\). We denote by \(\mathcal{B}^{+}_b(H)\) the subset of the non-negative functions in \(\mathcal{B}_b(H)\). Following \textit{M. Hairer} and \textit{J. C. Mattingly} [Ann. Math. (2) 164, No. 3, 993--1032 (2006; Zbl 1130.37038)], the Markov transition \((P_t)\) on a Polish space \(X\) is asymptotically strong Feller if for every \(x \in X\) there exists a totally separating system of pseudo-metrics \((d_n)_{n \geq 1}\) for \(X\) and a sequence \(t_n>0\) such that \[\inf_{U \in \mathcal{U}_x} \limsup_{n \longrightarrow +\infty} \sup_{y \in U} \|P_{t_n}(x,.)-P_{t_n}(y,.)\|_{d_n}=0,\] where \(\mathcal{U}_x\) is the collection of all open sets containing \(x\) and \(t_n \rightarrow +\infty\) as \(n \rightarrow +\infty\). For \(f : H \longrightarrow \mathbb{R}\), we put \[|\nabla(f)(x)|=\limsup_{\|x-y\|_{H} \longrightarrow 0}\frac{|f(x)-f(y)|}{\big\|x-y\big\|_{H}},\] \(\big\|\nabla(f)(x)\big|_{\infty}=\sup_{x \in H}|\nabla(f)(x)|\) and \(\text{Lip}(H)=\big\{f : \longrightarrow \mathbb{R}, \big\|\nabla(f)(x)\big|_{\infty}<\infty \}.\) In this setting, the authors prove the following Theorem. Under the previous assumption and suppose \(\lambda_{N_0}>\frac{K_1+K_2}{2}\) for \(N_0\) defined in (4). Then, \((P_t)\) is asymptotically strong Feller and we have the following asymptotic \(\log\)-Harnack inequality, for \(t>0,\) and \(f \in \mathcal{B}^{+}_b(H)\), \[P_t\log(f)(y) \leq \log P_t(f)(x)+\delta \|x-y\|_{H}^2+e^{-\kappa t} \|x-y\|_{H}\big\|\nabla \log(f)\big\|_{\infty}, \] where \(\kappa= \lambda_{N_0}-\frac{K_1+K_2}{2}\) and \(\delta=\frac{\lambda_{N_0}^2 \eta^2}{4 \kappa}.\) We further have the following asymptotic heat kernel estimate, for any \(f \in Lip(H) \cap \mathcal{B}^{+}_b(H)\), Moreover, if the embedding \(V \subset H\) is compact and \(\beta_2 < 2\lambda_1\) then, \((P_t)\) is ergodic, that is, there exists a unique invariant measure \(\mu\) for transition semigroup \(P_t\). The authors prove the asymptotically strong Feller property and the following asymptotic heat kernel estimate. For the ergodic property, according the asymptotic heat kernel estimate, the proof is accomplished by applying Krylov-Bogoliubov procedure.
scientific article; zbMATH DE number 7259056
scientific article