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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.
##### MSC:
 60H15 Stochastic partial differential equations (aspects of stochastic analysis) 37A25 Ergodicity, mixing, rates of mixing
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