Renormalization of stochastic nonlinear heat and wave equations driven by subordinate cylindrical Brownian noises

TL;DR

This paper develops a renormalization approach for stochastic nonlinear heat and wave equations driven by jump-type cylindrical Brownian noises, establishing local well-posedness results with different regularity properties.

Contribution

It introduces a novel renormalization method for jump-type cylindrical noises and analyzes the well-posedness of SNLH and SNLW with singular noises on a 2D torus.

Findings

SNLH solutions lack time-continuity due to jump noise.

Local well-posedness for SNLH is limited to quadratic nonlinearities.

SNLW solutions are time-continuous and well-posed for polynomial nonlinearities.

Abstract

In this paper, we study the stochastic nonlinear heat equations (SNLH) and stochastic nonlinear wave equations (SNLW) on two-dimensional torus driven by a subordinate cylindrical Brownian noise, which we define by the time-derivative of a cylindrical Brownian motion subordinated to a nondecreasing cadlag stochastic process. To construct the solution, we introduce a suitable renormalization. For SNLH, we cannot expect the time-continuity for the solutions because the noise is jump-type. Moreover, due to the low time-integrability of the solutions, we could establish a local well-posedness result for SNLH only with a quadratic nonlinearity. On the other hand, for SNLW, the solutions have time-continuity and we can show the local well-posedness for general polynomial nonlinearities. Through this example, we can see that the heat case behaves worse than the wave case in the singular noise of jump-type cases.