Fractional one-sided measure theoretic second-order elliptic operators and applications to stochastic partial differential equations

TL;DR

This paper introduces fractional measure theoretic elliptic operators and a new stochastic process called W-Brownian motion, establishing regularity, spectral bounds, and applications to stochastic PDEs.

Contribution

It presents novel fractional elliptic operators on the torus, defines W-Brownian motion, and links the associated Cameron-Martin space to Sobolev spaces, with applications to stochastic PDEs.

Findings

Sharp bounds for eigenvalue growth rates.

Relationship between W-Brownian motion's Cameron-Martin space and Sobolev spaces.

Applications to stochastic partial differential equations.

Abstract

In this work we introduce and study fractional measure theoretic elliptic operators on the torus and a new stochastic process named W-Brownian motion. We establish some regularity and spectral results related to the operators cited above, more precisely, we were able to provide sharp bounds for the growth rate of eigenvalues to an associated eigenvalue problem. Moreover, we show how the Cameron-Martin space associated to the W-Brownian motion relates to Sobolev spaces connected with the elliptic operators mentioned above. Finally applications of the theory developed on stochastic partial differential equations are given.