Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

An Itô type formula for the additive stochastic heat equation

Carlo Bellingeri

Abstract

We use the theory of regularity structures to develop an Itô formula for $u$, the solution of the one dimensional stochastic heat equation driven by space-time white noise with periodic boundary conditions. In particular for any smooth enough function $\varphi$ we can express the random distribution $(\partial_t-\partial_{xx})\varphi(u)$ and the random field $\varphi(u)$ in terms of the reconstruction of some modelled distributions. The resulting objects are then identified with some classical constructions of stochastic calculus.

An Itô type formula for the additive stochastic heat equation

Abstract

We use the theory of regularity structures to develop an Itô formula for , the solution of the one dimensional stochastic heat equation driven by space-time white noise with periodic boundary conditions. In particular for any smooth enough function we can express the random distribution and the random field in terms of the reconstruction of some modelled distributions. The resulting objects are then identified with some classical constructions of stochastic calculus.

Paper Structure

This paper contains 14 sections, 29 theorems, 290 equations.

Table of Contents

  1. Introduction
  2. Hölder spaces and Malliavin calculus
  3. Regularity structures
  4. Algebraic construction
  5. Models on a regularity structure
  6. The BPHZ renormalisation and the BPHZ model
  7. Calculus on regularity structures
  8. Modelled distributions
  9. Operations with the stochastic heat equation
  10. Itô formula
  11. Pathwise Itô formulae
  12. Identification of the differential formula
  13. Identification of the integral formula
  14. Renormalisation constants

Key Result

Theorem 1.1

Let $\varphi$ be a function of class $C^{4}_b(\mathbb{R})$, the space of $C^4$ functions with all its derivatives bounded. Then for any test function $\psi\colon \mathbb{R}\times \mathbb{T} \to \mathbb{R}$ with supp $(\psi) \subset (0,T) \times \mathbb{T}$, one has the a.s. equality Moreover for any $(t,x)\in [0,T]\times \mathbb{T}$ we have a.s. where in both formulae $P'_s(y)= \partial_x P_s(x)

Theorems & Definitions (78)

  • Theorem 1.1: Integral and differential Itô formula
  • Remark 1.2
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Remark 3.3
  • Remark 3.4
  • Remark 3.5
  • Definition 3.6
  • Proposition 3.7
  • ...and 68 more