Non relativistic and ultra relativistic limits in 2d stochastic nonlinear damped Klein-Gordon equation

TL;DR

This paper investigates the behavior of the two-dimensional stochastic nonlinear damped Klein-Gordon equation under non-relativistic and ultra-relativistic limits, using energy and Strichartz estimate methods to analyze parameter dependence.

Contribution

It introduces two methods to analyze parameter dependence in the limits of the stochastic Klein-Gordon equation, enhancing understanding of its asymptotic behaviors.

Findings

Parameter dependence clarified using energy method

Parameter dependence clarified using Strichartz estimates

Framework established for analyzing relativistic limits in stochastic PDEs

Abstract

We study the non relativistic and ultra relativistic limits in the two-dimensional nonlinear damped Klein-Gordon equation driven by a space-time white noise on the torus. In order to take the limits, it is crucial to clarify the parameter dependence in the estimates of solution. In this paper we present two methods to confirm this parameter dependence. One is the classical, simple energy method. Another is the method via Strichartz estimates.