Tamed Feynman-Kac diffusion processes: Killing-branching intertwine

TL;DR

This paper explores the interplay of killing and branching mechanisms in Feynman-Kac diffusion processes, providing path-wise analysis and insights into potential shapes like double wells.

Contribution

It introduces a framework for understanding tamed Feynman-Kac diffusions with killing and branching, supported by computer-assisted path-wise arguments in nonlinear models.

Findings

Killing effects are often overcompensated by branching in Feynman-Kac processes.

Negative potential regions correspond to branching rates, influencing the process dynamics.

Path-wise arguments support the consistency of the killing-branching scenario in one-dimensional models.

Abstract

Relaxation to equilibrium of a drifted Brownian motion is quantified by a probability density function, whose main (multiplicative) entry is an inferred Feynman-Kac kernel of the Schr\"{o}dinger semigroup operator. Although seemingly devoid of a natural probabilistic significance (except for its explicit path integral definition), the pertinent kernel relaxes to equilibrium as well. The implicit Feynman-Kac potential ${\cal{V}}(x)$, continuous, confining and bounded from below, may take negative values. If positive, ${\cal{V}}(x)$ can be interpreted as the killing rate of the decaying diffusion process. In case of relaxing F-K kernels the killing effects are tamed (often overcompensated). The taming inavoidably appears in conjunction with the existence of the negativity subdomains of ${\cal{V}}(x)$ in $R$. If locally ${\cal{V}}(x) < 0$, its sign inversion $- {\cal{V}}(x)$ can be interpreted as the branching (cloning, alternatively bifurcation) rate in the course of the other wise free random motion. The arising killed diffusion processes with branching, we interpret as the possible path-wise background of tamed (relaxing) Feynman-Kac diffusions. We present acomputer-assisted path-wise arguments, towards a consistency of the killing/branching taming scenario, for a number of nonlinear model systems in one space dimension. Special attention is paid to Feynman-Kac potential shapes, presumed to be in the double well form, where an analytic access to eigenvalues and eigenfunctions is scarce beyond the semiclassical regime.