The Burgers equation driven by a stochastic measure

TL;DR

This paper investigates a class of one-dimensional stochastic equations, including Burgers and heat equations, establishing their existence, uniqueness, and analyzing the averaging principle under minimal assumptions on the stochastic measure.

Contribution

It introduces a framework for stochastic equations driven by a measure with only sigma-additivity in probability, extending classical results to broader stochastic influences.

Findings

Existence and uniqueness of solutions proved.

Averaging principle established for the class of equations.

Includes classical Burgers and heat equations as special cases.

Abstract

We study the class of one-dimensional equations driven by a stochastic measure $\mu$. For $\mu$ we assume only $\sigma$-additivity in probability. This class of equations include the Burgers equation and the heat equation. The existence and uniqueness of the solution are proved, and the averaging principle for the equation is studied.