A connection between the stochastic heat equation and fractional Brownian motion, and a simple proof of a result of Talagrand
Carl E Mueller, University of Rochester
Zhixin Wu, DePauw University
Abstract
We give a new representation of fractional Brownian motion with Hurst
parameter $Hleqfrac{1}{2}$ 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 Lévy.
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