Generalized space-time fractional stochastic kinetic equation

TL;DR

This paper investigates a class of nonlinear space-time fractional stochastic kinetic equations with Gaussian noise, extending stochastic heat equations by incorporating fractional derivatives, and analyzes their existence, uniqueness, and properties.

Contribution

It provides necessary conditions for solution existence and uniqueness, and explores key properties like regularity, moments, and stationarity for these fractional stochastic equations.

Findings

Established conditions for solution existence and uniqueness.

Analyzed path regularity and second moment behavior.

Studied stationarity in the linear additive noise case.

Abstract

In this paper, we study a class of nonlinear space-time fractional stochastic kinetic equations in $\mathbb{R}^d$ with Gaussian noise which is white in time and homogeneous in space. This type of equation constitutes an extension of the non-linear stochastic heat equation involving fractional derivative in time and fractional Laplacian in space. We give a necessary condition on the spatial covariance for the existence and uniqueness of the solution. We also study various properties of the solution: path regularity, the behavior of second moment and the stationarity in the case of linear additive noise.