One-sided measure theoretic elliptic operators and applications to SDEs driven by Gaussian white noise with atomic intensity

TL;DR

This paper develops a novel measure-theoretic elliptic operator framework on the torus, linking it to Gaussian processes and stochastic differential equations driven by atomic measures, expanding the understanding of such operators and processes.

Contribution

It introduces a new class of elliptic operators with atomic measures, characterizes associated Sobolev spaces, and constructs $W$-Brownian motion with applications to SDEs driven by Gaussian white noise.

Findings

Defined the operator $ abla_{W,V}$ and associated test function spaces.

Established $H_{W,V}( ext{T})$ as a Sobolev-type space with reproducing kernel properties.

Constructed $W$-Brownian motion as a Feller process with jumps subordinated to $W$.

Abstract

We define the operator $D^+_VD^-_W:=\Delta_{W,V}$ on the one-dimensional torus $\mathbb{T}$. Here, $W$ and $V$ are functions inducing (possibly atomic) positive Borel measures on $\mathbb{T}$, and the derivatives are generalized lateral derivatives. For the first time in this work, the space of test functions $C^{\infty}_{W,V}(\mathbb{T})$ emerges as the natural regularity space for solutions of the eigenproblem associated with $\Delta_{W,V}$. Moreover, these spaces are essential for characterizing the energetic space $H_{W,V}(\mathbb{T})$ as a Sobolev-type space. By observing that the Sobolev-type spaces $H_{W,V}(\mathbb{T})$ with additional Dirichlet conditions are reproducing kernel Hilbert spaces, we introduce the so-called $W$-Brownian bridges as mean-zero Gaussian processes with associated Cameron-Martin spaces derived from these spaces. This framework allows us to introduce $W$-Brownian motion as a Feller process with a two-parameter semigroup and c\`adl\`ag sample paths, whose jumps are subordinated to the jumps of $W$. We establish a deep connection between $W$-Brownian motion and these Sobolev-type spaces through their associated Cameron-Martin spaces. Finally, as applications of the developed theory, we demonstrate the existence and uniqueness of related deterministic and stochastic differential equations.