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Regularisation by regular noise

Máté Gerencsér

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 $α>1-1/(2H)$ is recovered for all non-integer $H>1$.

Regularisation by regular noise

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 is recovered for all non-integer .

Paper Structure

This paper contains 6 sections, 4 theorems, 2 equations.

Table of Contents

  1. Introduction
  2. Preparations
  3. Notation and basics
  4. Stochastic sewing lemma
  5. Higher order expansion of solutions
  6. Proof of Theorem \ref{['thm:main']}

Key Result

Theorem 1.1

Let $H\in[1/2,\infty)\setminus\mathbb{Z}$ and $b\in C^\alpha$, where Then eq:main has a unique strong solution up to time $1$.

Theorems & Definitions (10)

  • Theorem 1.1
  • Remark 1.2
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Remark 2.5