A connection between the stochastic heat equation and fractional Brownian motion, and a simple proof of a result of Talagrand

TL;DR

This paper introduces a new SPDE-based representation of fractional Brownian motion for Hurst parameter H≤1/2, enabling simpler proofs of its hitting and double point properties at critical dimensions.

Contribution

It provides a novel SPDE representation of fractional Brownian motion that leverages the Markov property and time reversal, simplifying existing proofs of key properties.

Findings

Fractional Brownian motion does not hit points in the critical dimension.

Fractional Brownian motion does not have double points in the critical dimension.

The new representation simplifies proofs of these properties.

Abstract

We give a new representation of fractional Brownian motion with Hurst parameter H<=1/2 using stochastic partial differential equations. This representation allows us to use the Markov property and time reversal, tools which are not usually available for fractional Brownian motion. We then give simple proofs that fractional Brownian motion does not hit points in the critical dimension, and that it does not have double points in the critical dimension. These facts were already known, but our proofs are quite simple and use some ideas of Levy.