Sweetest taboo processes

TL;DR

This paper investigates taboo processes, a class of constrained stochastic processes, in simple geometries, highlighting their properties and potential applications in modeling diffusive phenomena with boundary constraints.

Contribution

It provides a detailed analysis of taboo processes in various geometries, including the Gaussian behavior of the stochastic angle in a circular annulus, which is less explored in physics.

Findings

Gaussian behavior of the stochastic angle in a circular annulus

Analysis of taboo processes in 1D, 2D, and 3D geometries

Comparison with reflecting stochastic differential equations

Abstract

Brownian dynamics play a key role in understanding the diffusive transport of micro particles in a bounded environment. In geometries containing confining walls, physical laws determine the behavior of the random trajectories at the boundaries. For impenetrable walls, imposing reflecting boundary conditions to the Brownian particles leads to dynamics described by reflecting stochastic differential equations. In practice, these stochastic differential equations as well as their refinements are quite challenging to handle, and more importantly, many physical processes are better modeled by processes conditioned to stay in a prescribed bounded region. In the mathematical literature, these processes are known as taboo processes, and despite their simplicity, at least compared to the reflecting stochastic differential equations approach, are surprisingly not much exploited in physics. This paper explores some aspect of taboo processes and other constrained processes in simple geometries: Interval in one dimension, circular annulus in two dimensions, hollow sphere in three dimensions, and more. In particular, for the two-dimensional taboo process in a circular annulus, the Gaussian behavior of the stochastic angle is established.