Partially Reflected Brownian Motion: A Stochastic Approach to Transport Phenomena

TL;DR

This paper explores how partially reflected Brownian motion models diffusive transport near semi-permeable interfaces, linking stochastic processes with Laplacian transport phenomena across various scientific fields.

Contribution

It provides an overview of the mathematical foundations and applications of partially reflected Brownian motion in modeling transport phenomena, emphasizing the role of the Dirichlet-to-Neumann operator.

Findings

Illustrates the connection between stochastic processes and Laplacian transport.

Highlights the use of local time and harmonic measure in modeling.

Discusses practical implications and future research directions.

Abstract

Transport phenomena are ubiquitous in nature and known to be important for various scientific domains. Examples can be found in physics, electrochemistry, heterogeneous catalysis, physiology, etc. To obtain new information about diffusive or Laplacian transport towards a semi-permeable or resistive interface, one can study the random trajectories of diffusing particles modeled, in a first approximation, by the partially reflected Brownian motion. This stochastic process turns out to be a convenient mathematical foundation for discrete, semi-continuous and continuous theoretical descriptions of diffusive transport. This paper presents an overview of these topics with a special emphasis on the close relation between stochastic processes with partial reflections and Laplacian transport phenomena. We give selected examples of these phenomena followed by a brief introduction to the partially reflected Brownian motion and related probabilistic topics (e.g., local time process and spread harmonic measure). A particular attention is paid to the use of the Dirichlet-to-Neumann operator. Some practical consequences and further perspectives are discussed.