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Optimal rate of convergence for approximations of SPDEs with non-regular drift

Oleg Butkovsky, Konstantinos Dareiotis, Máté Gerencsér

TL;DR

The optimal strong rate of convergence is proved without posing any regularity assumption on the non-linear reaction term and the proof relies on stochastic sewing techniques.

Abstract

A fully discrete finite difference scheme for stochastic reaction-diffusion equations driven by a $1+1$-dimensional white noise is studied. The optimal strong rate of convergence is proved without posing any regularity assumption on the non-linear reaction term. The proof relies on stochastic sewing techniques.

Optimal rate of convergence for approximations of SPDEs with non-regular drift

TL;DR

The optimal strong rate of convergence is proved without posing any regularity assumption on the non-linear reaction term and the proof relies on stochastic sewing techniques.

Abstract

A fully discrete finite difference scheme for stochastic reaction-diffusion equations driven by a -dimensional white noise is studied. The optimal strong rate of convergence is proved without posing any regularity assumption on the non-linear reaction term. The proof relies on stochastic sewing techniques.

Paper Structure

This paper contains 13 sections, 23 theorems, 4 equations.

Table of Contents

  1. Introduction
  2. Literature
  3. Notation
  4. Formulation
  5. Estimates on heat kernels and stochastic convolutions
  6. Definitions
  7. Discrete and continuous heat kernel bounds
  8. Discrete and continuous Ornstein-Uhlenbeck processes
  9. The sewing strategy
  10. Overview
  11. Estimate for $\mathcal{E}^{n,1}$
  12. Estimate for $\mathcal{E}^{n,2 }$
  13. Proof of Theorem \ref{['thm:main-theorem']}

Key Result

Theorem 1.0.1

For any $\varepsilon\in(0,1/2)$, bounded and measurable $b$, and any initial condition of class $\mathcal{C}^{1/2-\varepsilon}(\mathbb{T})$, the forward Euler finite difference approximation of eq:main in classical form converges strongly with rate $1/4-\varepsilon/2$ w.r.t. time and $1/2-\varepsilo

Theorems & Definitions (53)

  • Theorem 1.0.1
  • Remark 1.3.1
  • Theorem 1.3.2
  • Remark 1.3.3
  • Remark 1.3.4
  • Remark 1.3.5
  • Definition 2.1.1
  • Proposition 2.1.2
  • Remark 2.1.3
  • proof
  • ...and 43 more