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Asymptotic log-Harnack inequality and applications for SPDE with degenerate multiplicative noise. (English) Zbl 07259056
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 \[P_Nu :=\sum_{|k| \leq N}u_k e_k ,~~~~ u \in H .\] The authors consider the following stochastic evolution equation. \[du(t)=[-Au(t)+F(u(t))]dt+B(u(t))dW(t),~~~~~~u(0)=x \in H, ~~~~~~~(1.1)\] 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. 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 \[ 2 \langle -Au+F(u),u \rangle +\big\|B(u)\big\|_{L_2}^2 \leq \beta_1 + \beta_2 \big\|u\big\|_{H}^2-2 \big\|u\big\|_{V}^2.~~~(2.2)\]
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 \(\mathbb{P}.a.s.\) \[u(t)=x+\int_{0}^{t}\big(-Au(s)+F(u(s))\big) ds +\int_{0}^{t}B(u(s)) dW(s), ~~t \in [0,T].\]
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 \[P_tf(x)=\mathbb{E}(u^{x}(t)), f \in \mathcal{B}_b(H),\] \(\mathcal{B}_b(H)\) 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 M. Hairer and 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 proved 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 proof of the asymptotically strong Feller and the following asymptotic heat kernel estimate is in [W. Hong et al., Stat. Probab. Lett. 164, Article ID 108810, 7 p. (2020; Zbl 07259056)]. For the ergodic property, according the asymptotic heat kernel estimate, the proof is accomplished by applying Krylov-Bogoliubov procedure.
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
37A25 Ergodicity, mixing, rates of mixing
Full Text: DOI
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