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Elastic Brownian motion with random jumps from the boundary

Fausto Colantoni, Mirko D'Ovidio

TL;DR

This work develops a rigorous framework for elastic Brownian motion in a bounded domain where boundary hits trigger interior restarts according to a fixed measure, modeled by an SDE with boundary local time and a Poisson boundary-jump term. It provides a complete characterization through the generator, a pathwise construction, an invariant measure, and a spectral representation, including a Volterra equation that governs the boundary mass c(t) and a probabilistic representation of the solution. The paper also analyzes the trace process on the upper half-space, connects to subordination and Marchaud-type derivatives, and demonstrates an application to preventing trap-induced divergence of search times in narrow-neck domains. Together, these results offer a comprehensive probabilistic and analytic treatment of nonlocal Robin-type boundary conditions with resets, with potential implications for stochastic resetting, transport in bottleneck geometries, and diffusion processes with nonlocal boundary interactions.

Abstract

In this paper, we study elastic Brownian motion on a \(C^2\) domain. Instead of being killed at the boundary, the process restarts from a random position inside the domain. We characterize this process through its stochastic differential equation (SDE), its generator, and a description of the paths. We also derive the invariant probability measure and the spectral representation. At the end, we focus on the harmonic functions on the upper half-space to study the trace process.

Elastic Brownian motion with random jumps from the boundary

TL;DR

This work develops a rigorous framework for elastic Brownian motion in a bounded domain where boundary hits trigger interior restarts according to a fixed measure, modeled by an SDE with boundary local time and a Poisson boundary-jump term. It provides a complete characterization through the generator, a pathwise construction, an invariant measure, and a spectral representation, including a Volterra equation that governs the boundary mass c(t) and a probabilistic representation of the solution. The paper also analyzes the trace process on the upper half-space, connects to subordination and Marchaud-type derivatives, and demonstrates an application to preventing trap-induced divergence of search times in narrow-neck domains. Together, these results offer a comprehensive probabilistic and analytic treatment of nonlocal Robin-type boundary conditions with resets, with potential implications for stochastic resetting, transport in bottleneck geometries, and diffusion processes with nonlocal boundary interactions.

Abstract

In this paper, we study elastic Brownian motion on a domain. Instead of being killed at the boundary, the process restarts from a random position inside the domain. We characterize this process through its stochastic differential equation (SDE), its generator, and a description of the paths. We also derive the invariant probability measure and the spectral representation. At the end, we focus on the harmonic functions on the upper half-space to study the trace process.

Paper Structure

This paper contains 9 sections, 7 theorems, 152 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Stochastic Differential Equation
  3. Existence and Uniqueness
  4. Generator
  5. Paths description
  6. Invariant measure
  7. Spectral theorem
  8. Trace process on the upper half-space
  9. Escaping narrow-neck traps by jumps

Key Result

Theorem 1

Let the assumptions above hold. Then the SDE eq:SDE admits a pathwise unique strong solution $(X_t,L_t)_{t \ge 0}$, adapted to the filtration generated by $(B,\nu)$, with $X$ càdlàg and $L$ continuous and nondecreasing.

Figures (3)

  • Figure 1: A possible path for $X$ in a disk. On the left, the paths leading up to the first jump are displayed. On the right, a possible path for $X$ after the fourth jump, with the red points indicating restart points that follow a standard normal random variable.
  • Figure 2: Paths of the process $(W,X)$ on $H$. Left: after the first jump; Right: after the fourth jump. Red points indicate the boundary position from which the jumps start.
  • Figure 3: A possible dumbbell domain.

Theorems & Definitions (20)

  • Theorem 1
  • proof
  • Remark 1
  • Lemma 1: Concatenation of elastic reflected Brownian motions
  • proof
  • Remark 2
  • Theorem 2
  • proof
  • Remark 3
  • Theorem 3: Invariant measure
  • ...and 10 more