Propagation of Singularities for the Damped Stochastic Klein-Gordon Equation

TL;DR

This paper demonstrates that singularities in the 1+1 dimensional damped stochastic Klein-Gordon equation propagate similarly to the stochastic wave equation, revealing connections to microlocal analysis and wavefront set descriptions.

Contribution

It establishes the propagation of singularities for the damped stochastic Klein-Gordon equation with novel proofs that differ from those used for the wave equation.

Findings

Random singularities exist and propagate as in the stochastic wave equation.

Proof techniques are significantly different and more intuitive from the PDE perspective.

Results for the critically damped case imply those for the general case.

Abstract

For the $1+1$ dimensional damped stochastic Klein-Gordon equation, we show that random singularities associated with the law of the iterated logarithm exist and propogate in the same way as the stochastic wave equation. This provides evidence for possible connections to microlocal analysis, ie. the exact regularity and singularities described in this paper should admit wavefront set type descriptions whose propagation is determined by the highest order terms of the linear operator. Despite the results being exactly the same as those of the wave equation, our proofs are significantly different than the proofs for the wave equation. Miraculously, proving our results for the critically damped equation implies them for the general equation, which significantly simplifies the problem. Even after this simplification, many important parts of the proof are significantly different than (and we think are more intuitive from the PDE viewpoint compared to) existing proofs for the wave equation.