Intermittency and stochastic pseudo-differential equation with spatially inhomogeneous white noise
type: Item
instance of: scholarly article
title: Intermittency and stochastic pseudo-differential equation with spatially inhomogeneous white noise
zbMATH Open document ID: 1433.60030
published in: NoDEA. Nonlinear Differential Equations and Applications
publication date: 2019-02-28T00:00:00Z
review text: Consider the nonlinear stochastic partial differential equation in (1+1)-dimension \[ \bigl(\partial/\partial t+q(x,D_x)\bigr)u(t,x)=g(u(t,x))\bigl(\partial^2/\partial t\,\partial x\bigr)w_\rho(t,x) \] with initial condition \(u(0,x)=u_0(x)\), where \(q(x,D_x)\) is a pseudo-differential operator generating a stable-like process, and \(\partial^2w_\rho/\partial t\,\partial x\) is a space-time white noise with intensity measure \(dt\,\rho(dx)\). The authors first prove under some conditions the existence, uniqueness, and Hölder continuity of a mild solution. Then, and this is the main result, they study the intermittency of the solution. More precisely, by studying the moments of \(u(t,x)\), they obtain the existence of a smallest integer \(p\) such that \[ \inf_x\Bigl(\liminf_{t\to\infty}\frac{1}{t}\log\mathbb{E}\bigl[|u(t,x)|^p\bigr]\Bigr)0. \]
reviewed by: Jean Picard
Mathematics Subject Classification ID: 60H07
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 60G35
zbMATH DE Number: 7031082
zbMATH Keywords: stochastic pseudo-differential equation
zbMATH Keywords: spatially inhomogeneous white noise
zbMATH Keywords: intermittency
author: Junfeng Liu
MaRDI profile type: Publication
full work available at URL: s00030-018-0548-0
OpenAlex ID: W2907149452
cites work: Uniqueness in law for pure jump Markov processes
cites work: The stochastic heat equation: Feynman-Kac formula and intermittence.
cites work: A super-Brownian motion with a single point catalyst
cites work: Intermittence and nonlinear parabolic stochastic partial differential equations
cites work: Remarks on non-linear noise excitability of some stochastic heat equations
cites work: Moment bounds for a class of fractional stochastic heat equations
cites work: Non-linear noise excitation for some space-time fractional stochastic equations in bounded domains
cites work: Asymptotic properties of some space-time fractional stochastic equations
cites work: Solving the KPZ equation
cites work: Stochastic heat equations with general multiplicative Gaussian noises: Hölder continuity and intermittency
cites work: Pseudodifferential operators with variable order of differentiation generating Feller semigroups
cites work: Solving a non-linear stochastic pseudo-differential equation of Burgers type
cites work: Stochastic generalized Burgers equations driven by fractional noises
cites work: Q5166655
cites work: Nonlinear noise excitation of intermittent stochastic PDEs and the topology of LCA groups
cites work: On Markov process generated by pseudodifferential operator of variable order
cites work: Symmetric Stable Laws and Stable-Like Jump-Diffusions
cites work: Markov processes associated with certain integro-differential operators
cites work: Continuity estimates for solutions of parabolic equations associated with jump type Dirichlet forms
cites work: Uniform estimates for fundamental solutions associated with nonlocal Dirichlet forms
cites work: Stochastic partial differential equations for some measure-valued diffusions
cites work: On a nonlinear stochastic pseudo-differential equation driven by fractional noise
cites work: Generalized Anderson model with time-space multiplicative fractional noise
cites work: Stable-like processes: Construction of the transition density and the behavior of sample paths near \(t=0\)
cites work: Pathwise uniqueness of the stochastic heat equation with spatially inhomogeneous white noise
cites work: Recent developments on stochastic heat equation with additive fractional-colored noise
cites work: Q3747436
cites work: Intermittency for stochastic partial differential equations driven by strongly inhomogeneous space-time white noises
cites work: Heat equation with strongly inhomogeneous noise
cites work: Space-time regularity of catalytic super-Brownian motion
Wikidata QID: Q128640866
DOI: 10.1007/S00030-018-0548-0
description: scientific article; zbMATH DE number 7031082