On a stochastic nonlinear equation arising from 1D integro-differential scalar conservation laws

Abstract

In this paper we study the initial problem for a stochastic nonlinear equation arising from 1D integrodifferential scalar conservation laws. The equation is driven by Lévy space-time white noise in the following form:where A is an integro-differential operator associated with a symmetric, nonlocal, regular Dirichlet form, and F t,x stands for a Lévy space-time white noise. The problem is interpreted as a stochastic integral equation of jump type involving certain convolution kernels. Existence of a unique local (in time) L 2 (R)-valued solution is obtained.

Decaying turbulence for the fractional subcritical Burgers equation

“…The fractional Burgers equation has also been considered in a variety of other settings. For a probabilistic interpretation of (2), see the papers of Jourdain, Méléard and Woyczynski [29], Jourdain and Roux [30] and Truman and Wu [39]. For a proof of ergodicity for the fractional Burgers equation with space-time white noise, see the paper of Brzezniak, Debbi and Goldys [12].…”

Ergodicity of linear SPDE driven by Lévy noise

“…There are a great amount of literature on this topic, for example, [1][2][3][4][5][6][7][8][9] and references therein. However, more recently, stochastic evolution equations and stochastic differential equations with the perturbations of Poisson noise or Lévy noise have attracted more and more attentions, for instances, [10][11][12][13][14][15][16][17][18], etc. For example, the existence and uniqueness of the strong solution for the following SPDE…”

“…whereṄ is Lévy noise was studied in [11,[16][17]. In this article, we will study the ergodic behavior of a class of linear partial differential equations driven by Lévy noise.…”

Regularity of a fractional partial differential equation driven by space-time white noise

“…In fact, these kinds of equations can be widely used in physics, fractal medium, quantum fields, risk management and other fields; see [4,9,12]. This paper aims to give a deep study of the work introduced by L. Debbi and M. Dozzi in 2005 [4].…”