Tamed Feynman–Kac diffusion processes: killing-branching intertwine

Abstract

Relaxation to equilibrium of a drifted Brownian motion can be quantified by a transition probability density function, whose main multiplicative (Doob-like weighted) entry is an inferred Feynman–Kac kernel of the Schrödinger semigroup operator. The pertinent kernel captures a complete information about the time evolution and actually controls the asymptotic equilibration. The implicit Feynman–Kac potential

, which is confining, continuous and bounded from below, may take negative values. If positive,

V ( x ) $\mathcal{V}\left(x\right)$

can be considered as the killing rate of the decaying diffusion process. In the case of relaxing diffusion, killing effects need to be tamed. For unbounded random motion, the taming unavoidably appears in conjunction with the existence of the negativity subdomains of

V ( x ) $\mathcal{V}\left(x\right)$

in R . If locally

V ( x ) < 0 $\mathcal{V}\left(x\right){< }0$

, we assign a probabilistic meaning to the sign inverted potential

− V ( x ) $-\mathcal{V}\left(x\right)$

, and interpret it as the branching (cloning, alternatively trajectory bifurcation) rate. This points towards the killed diffusion with branching, as a possible path-wise background for the time evolution of the considered Feynman–Kac kernel. The emergent tamed Feynman–Kac diffusion scenario is discussed in detail for the exemplary quadratic potential

V ( x ) $\mathcal{V}\left(x\right)$

. The killing/branching algorithm, introduced in the present paper, induces a statistical ensemble of random paths, whose time evolution and asymptotics prove to be consistent with the exact analytical outcomes. This sets a rationale for a subsequent computer-assisted path-wise analysis of the validity of the tamed F–K diffusion concept, for a number of nonlinear models in one space dimension, where analytic results are scarce. Special attention is paid to Feynman–Kac potential shapes in the double well form, where an analytic access to eigenvalues and eigenfunctions of the related Schrödinger semigroup generator is unavailable beyond the semiclassical (or perturbative) regime. Throughout the paper the dynamics refers to the real time t ≥ 0. Since the Newton-type equations of motion for admissible classical trajectories have a Euclidean form (due to the sign inverted force term), we give a brief resume of a couple of their explicit solutions, without recourse to “imaginary” (Euclidean) time intuitions, and the instanton lore of related quantum model systems.