Planar Diffusions with Rank-Based Characteristics: Transition Probabilities, Time Reversal, Maximality and Perturbed Tanaka equations

TL;DR

This paper constructs and analyzes a planar diffusion process with rank-based characteristics, computing transition probabilities, exploring time reversal, and connecting it to perturbed Tanaka equations and skew Brownian motion.

Contribution

It introduces a novel rank-dependent planar diffusion model and provides detailed analysis of its transition probabilities, time reversal, and connections to local time and skew Brownian motion.

Findings

Explicit transition probabilities derived

Time reversal involves singular components governed by local time

Representation in terms of bang-bang drift and local time

Abstract

We construct a planar diffusion process whose infinitesimal generator depends only on the order of the components of the process. Speaking informally and a bit imprecisely for the moment, imagine you run two Brownian-like particles on the real line. At any given time, you assign positive drift g and diffusion {\sigma} to the laggard; and you assign negative drift -h and diffusion {\rho} to the leader. We compute the transition probabilities of this process, discuss its realization in terms of appropriate systems of stochastic differential equations, study its dynamics under a time reversal, and note that these involve singularly continuous components governed by local time. Crucial in our analysis are properties of Brownian and semimartingale local time; properties of the generalized perturbed Tanaka equation which we study here in detail; and those of a one-dimensional diffusion with bang-bang drift. We also show that our planar diffusion can be represented in terms of a process with bang-bang drift, its local time at the origin, and an independent standard Brownian motion, in a form which can be construed as a two-dimensional analogue of the stochastic equation satisfied by the so-called skew Brownian motion.