Time continuity of weak-predictable random field solutions

TL;DR

This paper investigates the time continuity properties of weak-predictable solutions to jump-discontinuous heat equations driven by Poisson noise, establishing their mean and mean-square continuity over any time interval.

Contribution

It proves that such solutions possess continuous modifications in time, which is crucial for understanding their global existence or non-existence.

Findings

Solutions are mean and mean-square continuous in time.

Solutions have continuous versions or modifications for any time interval.

Results apply to jump-discontinuous heat equations with Poisson noise.

Abstract

The question of global existence or non-existence of solution to a given stochastic partial differential equation under some non-linear conditions always comes to mind. To show that our weak-predictable random field solutions do not have global existence for all time , it requires that we first establish that the solutions exhibit continuity in time property. The results discuss the mean-square and mean continuity in time of a class of jump-discontinuous heat equations perturbed by compensated and non-compensated Poisson random noises respectively; and we showed that our mild solutions are mean and mean-square continuous in time for any time interval ; better put, our solutions have continuous versions or modifications for any time interval.