A multiparameter Garsia-Rodemich-Rumsey inequality and some applications

TL;DR

This paper generalizes the Garsia-Rodemich-Rumsey inequality to multiple parameters and applies it to establish joint Hölder continuity for fractional Brownian fields and solutions to stochastic heat equations with white noise.

Contribution

It introduces a multiparameter version of the Garsia-Rodemich-Rumsey inequality and demonstrates its applications in stochastic analysis.

Findings

Established joint Hölder continuity for fractional Brownian fields.

Proved Hölder regularity for solutions to stochastic heat equations.

Extended classical inequalities to multiparameter stochastic processes.

Abstract

We extend the classical Garsia-Rodemich-Rumsey inequality to the multiparameter situation. The new inequality is applied to obtain some joint H\"older continuity along the rectangles for fractional Brownian fields $W(t, x)$ and for the solution $u(t, y)$ of stochastic heat equation with additive white noise.