It\^o's formula for the $L_{p}$-norm of stochastic $W^{1}_{p}$-valued processes

TL;DR

This paper establishes Itô's formula for the $L_{p}$-norm of stochastic processes valued in the Sobolev space $W^{1}_{p}$, advancing the mathematical tools available for analyzing stochastic partial differential equations in divergence form.

Contribution

It provides the first proof of Itô's formula for the $L_{p}$-norm of stochastic $W^{1}_{p}$-valued processes in the context of SPDEs in divergence form.

Findings

Itô's formula is valid for $L_{p}$-norms of stochastic $W^{1}_{p}$-valued processes.

The result applies to solutions of SPDEs in divergence form.

This enhances analytical techniques for SPDEs in Sobolev spaces.

Abstract

We prove It\^o's formula for the $L_{p}$-norm of a stochastic $W^{1}_{p}$-valued processes appearing in the theory of SPDEs in divergence form.