Stochastic n-point D-bifurcations of stochastic L\'evy flows and their complexity on finite spaces

TL;DR

This paper extends the concept of stochastic D-bifurcations to n-point motions in stochastic flows, revealing new bifurcation behaviors and complexities on finite spaces beyond classical Brownian flows.

Contribution

It generalizes the notion of stochastic D-bifurcations to n-point motions and constructs minimal examples demonstrating the failure of rigidity in finite spaces.

Findings

Stochastic Brownian flows are rigid, with bifurcations only at levels 1 and 2.

General stochastic Markov semiflows can have bifurcations at higher levels.

The paper constructs minimal examples showing the complexity of n-point bifurcations.

Abstract

This article refines the classical notion of a stochastic D-bifurcation to the respective family of n-point motions for homogeneous Markovian stochastic semiflows, such as stochastic Brownian flows of homeomorphisms, and their generalizations. This notion essentially detects at which level $k\leq n$ the support of the invariant measure of the k-point bifurcation has more than one connected component. Stochastic Brownian flows and their invariant measures which were shown by Kunita (1990) to be rigid, in the sense of being uniquely determined by the $1$-and $2$-point motions, and hence only stochastic n-point bifurcation of level $n=1$ or $n=2$ can occur. For general homogeneous stochastic Markov semiflows this turns out to be false. This article constructs minimal examples of where this rigidity is false in general on finite space and studies the complexity of the resulting n-point bifurcations.