Riemann integral of a random function and the parabolic equation with a general stochastic measure

TL;DR

This paper investigates the weak solutions of stochastic parabolic equations driven by general stochastic measures, focusing on defining and analyzing the Riemann integral of random functions as limits of sums.

Contribution

It introduces a framework for the Riemann integral of random functions in stochastic PDEs with general stochastic measures, expanding the understanding of such integrals.

Findings

Weak solutions are obtained for stochastic parabolic equations with general stochastic measures.

Properties of the Riemann integral of random functions are established.

The integral is shown as a limit in probability of Riemann sums.

Abstract

For stochastic parabolic equation driven by a general stochastic measure, the weak solution is obtained. The integral of a random function in the equation is considered as a limit in probability of Riemann integral sums. Basic properties of such integrals are studied in the paper.