Asymptotic log-Harnack inequality and applications for SPDE with degenerate multiplicative noise

Hong, Wei; Li, Shihu; Liu, Wei

doi:10.1016/j.spl.2020.108810

Cited by 10 publications

(7 citation statements)

References 20 publications

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“…Step 3: This step is devoted to investigating the term 14), which is similar to the recent work [27], we include it here for the completeness.…”

Section: Weak Convergencementioning

confidence: 99%

“…Later on, such framework has been substantially generalized by the third named author and Röckner in [24,25,26] to more general hypothesises only fulfilling the local monotonicity and generalized coercivity, which covers several SPDEs such as the stochastic porous media equations, stochastic fast-diffusion equations, stochastic 2D Navier-Stokes equations and other hydrodynamical type models, stochastic p-Laplace equations, stochastic power law fluid equations, stochastic Ladyzhenskaya models, etc. We also refer the interested readers to [6,14,22,23,27,32,39,41,42] and reference therein for the properties of solutions associated with such framework.…”

Section: Introductionmentioning

confidence: 99%

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Freidlin-Wentzell Type Large Deviation Principle for Multi-Scale Locally Monotone SPDEs

Hong¹,

Li²,

Liu³

2021

Preprint

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This work is concerned with the Freidlin-Wentzell type large deviation principle for a family of multi-scale quasi-linear and semi-linear stochastic partial differential equations (SPDEs) with small multiplicative noise under the generalized variational setting, which extend several existing works to the multiscale process. Employing the weak convergence method developed by Dupuis and Ellis [7] and Khasminskii's time discretization approach [18], the Laplace principle for SPDEs will be derived, which is equivalent to the large deviation principle. In particular, in this paper, we do not assume any compactness of the embedding on the Gelfand triple we considered in order to deal with the case of bounded and unbounded domains in some concrete models. Our main results are applicable to a wide family of SPDEs such as stochastic porous media equations, stochastic fast-diffusion equations, stochastic 2D hydrodynamical type models, stochastic p-Laplace equations, stochastic power law fluid equations and stochastic Ladyzhenskaya models.

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“…Step 3: This step is devoted to investigating the term 14), which is similar to the recent work [27], we include it here for the completeness.…”

Section: Weak Convergencementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Freidlin-Wentzell Type Large Deviation Principle for Multi-Scale Locally Monotone SPDEs

Hong¹,

Li²,

Liu³

2021

Preprint

Self Cite

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“…where x ∈ H. Due to technical difficulties, we consider the cases n = 2, 3 and r ∈ [3, ∞) only. In the degenerate multiplicative noise case, the asymptotic log-Harnack inequality for several kinds of models on stochastic differential systems with infinite memory is established in [4] and for a class of semilinear SPDEs is obtained in [21]. Let L 2 (H) = L 2 (H, H) be the space of all Hilbert-Schmidt operators from H to H. We need the following assumptions on the noise coefficient to obtain our main results.…”

Section: The Scbf Equations Perturbed By Multiplicative Noisementioning

confidence: 99%

“…The asymptotic log-Harnack inequality and some of its consequent properties for a class of stochastic 2D hydrodynamical-type systems driven by degenerate noise are established in [20]. In [21], the authors established asymptotic log-Harnack inequality and discussed its applications for semilinear SPDEs with degenerate multiplicative noise by the coupling method. For a sample literature on the ergodic theory for the 2D stochastic Navier-Stokes equations subjected to degenerate noise, we refer the interested readers to [7,12,13,14,17], etc.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic log-Harnack inequality for the stochastic convective Brinkman-Forchheimer equations with degenerate noise

Mohan

2020

Preprint

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In this work, we consider the two and three dimensional stochastic convective Brinkman-Forchheimer (SCBF) equations and examine some asymptotic behaviors of its strong solution. We establish the asymptotic log-Harnack inequality for the transition semigroup associated with the SCBF equations driven by additive as well as multiplicative degenerate noise via the asymptotic coupling method. As applications of the asymptotic log-Harnack inequality, we derive the gradient estimate, asymptotic irreducibility, asymptotic strong Feller property, asymptotic heat kernel estimate and ergodicity. Whenever the absorption exponent r ∈ (3, ∞), the asymptotic log-Harnack inequality is obtained without any restriction on the Brinkman coefficient (effective viscosity) µ > 0, the Darcy coefficient α > 0 and the Forchheimer coefficient β > 0.

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“…the case p > 2, may be treated more easily with the Krylov-Bogoliubov method [14], as is done in the variational setup in [4]. Previous approaches include dimension-free Harnack inequalities [29,33,38] and the more direct approach via irreducibility [54]. In the case of multiplicative noise, existence of invariant measures for monotone drift equations has been proved in [17].…”

Section: Introductionmentioning

confidence: 99%

Stability and moment estimates for the stochastic singular $Φ$-Laplace equation

Seib¹,

Stannat²,

Tölle³

2021

Preprint

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We study stability, long-time behavior and moment estimates for stochastic evolution equations with additive Wiener noise and with singular drift given by a divergence type quasilinear diffusion operator which may not necessarily exhibit a homogeneous diffusivity. Our results cover the singular p-Laplace and, more generally, singular Φ-Laplace equations with zero Dirichlet boundary conditions. We obtain improved moment estimates and quantitative convergence rates of the ergodic semigroup to the unique invariant measure, classified in a systematic way according to the degree of local degeneracy of the potential at the origin. We obtain new concentration results for the invariant measure and establish maximal dissipativity of the associated Kolmogorov operator. In particular, we recover the results for the curve shortening flow in the plane by Es-Sarhir, von Renesse and Stannat, NoDEA 16(9), 2012.

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Asymptotic log-Harnack inequality and applications for SPDE with degenerate multiplicative noise

Cited by 10 publications

References 20 publications

Freidlin-Wentzell Type Large Deviation Principle for Multi-Scale Locally Monotone SPDEs

Freidlin-Wentzell Type Large Deviation Principle for Multi-Scale Locally Monotone SPDEs

Asymptotic log-Harnack inequality for the stochastic convective Brinkman-Forchheimer equations with degenerate noise

Stability and moment estimates for the stochastic singular $Φ$-Laplace equation

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