Tanaka's equation on the circle and stochastic flows

TL;DR

This paper extends Tanaka's equation to flows on a circle graph, classifies solutions via probability measures, and analyzes the structure and support of these flows, revealing complex behaviors depending on the measures.

Contribution

It introduces a classification of solutions to Tanaka's equation on a circle graph using pairs of probability measures and analyzes the structure of the resulting flows.

Findings

Flows of kernels are classified by pairs of probability measures with mean 1/2.

Supports of flows contain finitely many points, which can be arbitrarily large.

The behavior at vertices is governed by the associated probability measures.

Abstract

We define a Tanaka's equation on an oriented graph with two edges and two vertices. This graph will be embedded in the unit circle. Extending this equation to flows of kernels, we show that the laws of the flows of kernels $K$ solution of Tanaka's equation can be classified by pairs of probability measures $(m^+,m^-)$ on $[0,1]$, with mean 1/2. What happens at the first vertex is governed by $m^+$, and at the second by $m^-$. For each vertex $P$, we construct a sequence of stopping times along which the image of the whole circle by $K$ is reduced to $P$. We also prove that the supports of these flows contains a finite number of points, and that except for some particular cases this number of points can be arbitrarily large.