On local time for the solution to a white noise driven heat equation

TL;DR

This paper investigates the local time existence for solutions of certain Gaussian processes arising from stochastic heat equations driven by white noise, demonstrating their properties as Gaussian integrators on finite intervals.

Contribution

It establishes the existence of local time for solutions to stochastic heat equations driven by white noise, using the framework of Gaussian integrators generated by invertible operators.

Findings

Solutions have local time on finite intervals with respect to space.

Processes are Gaussian integrators generated by invertible operators.

Local time exists for the class of processes considered.

Abstract

In this article we discuss the existence of local time for a class of Gaussian processes which appears as the solutions to some stochastic evolution equations. We show that on small intervals such processes are Gaussian integrators generated by a continuously invertible operators. This allows us to conclude that the considered processes have a local time on any finite interval with respect to spatial variable.