Rescaling nonlinear noise for 1D stochastic parabolic equations

TL;DR

This paper demonstrates that rescaling noise at extremum points in 1D stochastic parabolic equations causes solutions to converge to a space-independent martingale, revealing new insights into the noise's influence on solution behavior.

Contribution

It introduces a novel noise rescaling technique at extremum points and shows its effect on the convergence of solutions to a martingale in 1D stochastic parabolic equations.

Findings

Solutions converge to a space-independent martingale after noise rescaling.

Rescaling at extremum points alters the solution's long-term behavior.

The approach provides new understanding of noise effects in stochastic PDEs.

Abstract

In this paper we will show that the solution of 1D stochastic parabolic equation with additive noise converges to a martingale (independent upon space variable) when we rescale noise at the extremum points of the process.