On ill-posedness of nonlinear stochastic wave equations driven by rough noise

TL;DR

This paper demonstrates that nonlinear stochastic wave equations driven by very rough fractional noise are fundamentally ill-posed, indicating limits to defining solutions as the noise's roughness increases beyond a threshold.

Contribution

It establishes a fundamental ill-posedness threshold for nonlinear stochastic wave equations driven by fractional noise, extending previous results and clarifying the limits of solution interpretability.

Findings

Ill-posedness occurs when noise roughness exceeds a critical threshold.

No meaningful interpretation of the model is possible with sufficiently rough noise.

Results extend and clarify earlier findings on fractional noise-driven equations.

Abstract

We highlight a fundamental ill-posedness issue for nonlinear stochastic wave equations driven by a fractional noise. Namely, if the noise becomes too rough (i.e., the sum of its Hurst indexes becomes too small), then there is essentially no hope to provide an interpretation of the model, whether directly or through a Wick-type renormalization procedure. This phenomenon can be compared with the situation of a general SDE driven by a two-dimensional fractional noise of index $H\leq \frac{1}{4}$. Our results clarify and extend previous similar properties exhibited in [3] or in [14].