×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bao, J.; Wang, F.-Y.; Yuan, C., Asymptotic log-harnack inequality and applications for stochastic systems of infinite memory, Stochastic Process. Appl., 129, 4576-4596 (2019) · Zbl 1433.60032
[2] Brzezniak, Z.; Liu, W.; Zhu, J., Strong solutions for SPDE with locally monotone coefficients driven by Lévy noise, Nonlinear Anal. RWA, 17, 283-310 (2014) · Zbl 1310.60091
[3] Butkovsky, O.; Kulik, A.; Scheutzow, M., Generalized couplings and ergodic rates for SPDEs and other Markov models, Ann. Appl. Probab., 30, 1-39 (2020) · Zbl 1434.60147
[4] Constantin, P.; Glatt-Holtz, N.; Vicol, V., Unique ergodicity for fractionally dissipated stochastically forced 2D Euler equations, Comm. Math. Phys., 330, 819-857 (2014) · Zbl 1294.35078
[5] E, W.; Mattingly, J. C., Ergodicity for the Navier-Stokes equation with degenerate random forcing: finite-dimensional approximation, Comm. Pure Appl. Math., 54, 1386-1402 (2001) · Zbl 1024.76012
[6] Földes, J.; Glatt-Holtz, N.; Richards, G.; Thomann, E., Ergodic and mixing properties of the boussinesq equations with a degenerate random forcing, J. Funct. Anal., 269, 2427-2504 (2015) · Zbl 1354.37056
[7] Hairer, M., Exponential mixing properties of stochastic PDEs through asymptotic coupling, Probab. Theory Related Fields, 124, 345-380 (2002) · Zbl 1032.60056
[8] Hairer, M.; Mattingly, J. C., Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing, Ann. of Math., 164, 993-1032 (2006) · Zbl 1130.37038
[9] Kulik, A.; Scheutzow, M., Generalized couplings and convergence of transition probabilities, Probab. Theory Related Fields, 171, 333-376 (2018) · Zbl 1392.60061
[10] Li, S.; Liu, W.; Xie, Y., Ergodicity of 3D leray-\( \alpha\) model with fractional dissipation and degenerate stochastic forcing, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 22, Article 1950002 pp. (2019), 20pp · Zbl 1447.60108
[11] Liu, W., Harnack inequality and applications for stochastic evolution equations with monotone drifts, J. Evol. Equations, 9, 747-770 (2009) · Zbl 1239.60058
[12] Liu, W.; Röckner, M., SPDE in Hilbert space with locally monotone coefficients, J. Funct. Anal., 259, 2902-2922 (2010) · Zbl 1236.60064
[13] Liu, W.; Röckner, M., Stochastic Partial Differential Equations: An Introduction (2015), Springer · Zbl 1361.60002
[14] Liu, W.; Wang, F.-Y., Harnack inequality and strong feller property for stochastic fast-diffusion equations, J. Math. Anal. Appl., 342, 651-662 (2008) · Zbl 1151.60032
[15] Odasso, C., Exponential mixing for stochastic PDEs: the non-additive case, Probab. Theory Related Fields, 140, 41-82 (2008) · Zbl 1137.60030
[16] Wang, F.-Y., Logarithmic Sobolev inequalities on noncompact Riemannian manifolds, Probab. Theory Related Fields, 109, 417-424 (1997) · Zbl 0887.35012
[17] Wang, F.-Y., Harnack inequality and applications for stochastic generalized porous media equations, Ann. Probab., 35, 1333-1350 (2007) · Zbl 1129.60060
[18] Wang, F.-Y., Harnack inequality for SDE with multiplicative noise and extension to Neumann semigroup on nonconvex manifolds, Ann. Probab., 39, 1449-1467 (2011) · Zbl 1238.60069
[19] Wang, F.-Y., Harnack Inequalities and Applications for Stochastic Partial Differential Equations (2013), Springer: Springer Berlin
[20] Wang, F.-Y.; Zhang, T. S., Log-harnack inequality for mild solutions of SPDEs with multiplicative noise, Stochastic Process. Appl., 124, 1261-1274 (2014) · Zbl 1285.60065
[21] Xu, L., A modified log-harnack inequality and asymptotically strong Feller property, J. Evol. Equations, 11, 925-942 (2011) · Zbl 1270.60085
[22] Zhang, S.-Q., Harnack inequality for semilinear SPDE with multiplicative noise, Statist. Probab. Lett., 83, 1184-1192 (2013) · Zbl 1267.60075
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.