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

Alexandre B. Simas, Kelvin J. R. Sousa

TL;DR

The paper develops a unified framework for one-sided elliptic operators on the torus driven by irregular measures induced by strictly increasing cadlag/caglad functions $W$ and $V$. It introduces $D^-_W$ and $D^+_V$ as generalized lateral derivatives, defines the $W$-$V$-Sobolev spaces $H_{W,V}(\mathbb{T})$, and proves a regularity theory including a dense space of smooth functions $C^{\infty}_{W,V}(\mathbb{T})$ and sharp eigenfunction regularity in $C^{\infty}_{W,V,0}(\mathbb{T})$. The construction of $W$-Brownian bridge and $W$-Brownian motion, together with their Cameron–Martin spaces and a two-parameter Feller semigroup, provides a probabilistic backbone for stochastic analysis in this generalized setting. These tools enable existence and uniqueness results for deterministic and stochastic equations, including a Matérn-like SPDE driven by pathwise Gaussian white noise $\dot{B}_W$ in $H^{-1}_{W,V,\mathcal{D}}(\mathbb{T})$. Overall, the work connects generalized differential operators with Gaussian processes under atomic intensity and offers a tractable route to SDE/SPDE analysis in irregular media.

Abstract

We define the operator $D^+_VD^-_W:=Δ_{W,V}$ 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 $Δ_{W,V}$. 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 càdlàg 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.

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

TL;DR

The paper develops a unified framework for one-sided elliptic operators on the torus driven by irregular measures induced by strictly increasing cadlag/caglad functions and . It introduces and as generalized lateral derivatives, defines the --Sobolev spaces , and proves a regularity theory including a dense space of smooth functions and sharp eigenfunction regularity in . The construction of -Brownian bridge and -Brownian motion, together with their Cameron–Martin spaces and a two-parameter Feller semigroup, provides a probabilistic backbone for stochastic analysis in this generalized setting. These tools enable existence and uniqueness results for deterministic and stochastic equations, including a Matérn-like SPDE driven by pathwise Gaussian white noise in . Overall, the work connects generalized differential operators with Gaussian processes under atomic intensity and offers a tractable route to SDE/SPDE analysis in irregular media.

Abstract

We define the operator on the one-dimensional torus . Here, and are functions inducing (possibly atomic) positive Borel measures on , and the derivatives are generalized lateral derivatives. For the first time in this work, the space of test functions emerges as the natural regularity space for solutions of the eigenproblem associated with . Moreover, these spaces are essential for characterizing the energetic space as a Sobolev-type space. By observing that the Sobolev-type spaces with additional Dirichlet conditions are reproducing kernel Hilbert spaces, we introduce the so-called -Brownian bridges as mean-zero Gaussian processes with associated Cameron-Martin spaces derived from these spaces. This framework allows us to introduce -Brownian motion as a Feller process with a two-parameter semigroup and càdlàg sample paths, whose jumps are subordinated to the jumps of . We establish a deep connection between -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.

Paper Structure

This paper contains 8 sections, 32 theorems, 159 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The one-sided differential operator
  3. $W$-$V$-Sobolev Spaces
  4. The Space $C^{\infty}_{W,V}(\mathbb{T})$
  5. $W$-$V$-Sobolev spaces and weak derivatives
  6. One-sided second-order elliptic operators
  7. $W$-Brownian bridge and $W$-Brownian motion
  8. Applications to stochastic differential equations

Key Result

Lemma 1

If $f : \mathbb{T}\to\mathbb{R}$ is càglàd and $D^{+}_{V}f\equiv 0$, then $f$ is a constant function. Similarly, if $g : \mathbb{T}\to\mathbb{R}$ is càdlàg and $D^{-}_{W}g\equiv 0$, then $g$ is a constant function.

Figures (1)

  • Figure 1: An example of a realization of a $W$-Brownian bridge (middle) and of a $W$-Brownian motion (right) for a specific function $W$ (left).

Theorems & Definitions (103)

  • Definition 1
  • Definition 2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Remark 1
  • Remark 2
  • Theorem 1
  • proof
  • ...and 93 more