Regularisation by regular noise

TL;DR

This paper demonstrates that adding smooth fractional Brownian noise to ill-posed differential equations can restore well-posedness, even when the noise is more regular than the original drift, extending known regularity conditions.

Contribution

It introduces a novel approach of using fractional Brownian noise to regularize ill-posed differential equations, broadening the understanding of noise-induced well-posedness.

Findings

Regular noise restores well-posedness in ill-posed equations.

The regularity condition $ ext{alpha}>1-1/(2H)$ is confirmed for all non-integer $H>1$.

Fractional Brownian noise can be more regular than the drift component.

Abstract

We show that perturbing ill-posed differential equations with (potentially very) smooth random processes can restore well-posedness -- even if the perturbation is (potentially much) more regular than the drift component of the solution. The noise considered is of fractional Brownian type, and the familiar regularity condition $\alpha>1-1/(2H)$ is recovered for all non-integer $H>1$.