Killing versus branching: Unexplored facets of diffusive relaxation.

We analyze the relaxation dynamics of Feynman-Kac path integral kernel functions, in terms of branching diffusion processes with killing. This amounts to the killing versus branching approach to path integration, which seems to be a novelty in the pathwise description of conditioned Brownian motions and diffusion processes with absorbing boundaries. There, Feynman-Kac kernels appear as building blocks of inferred (Fokker-Planck) transition probability density functions.

Fractional Laplacians in bounded domains: Killed, reflected, censored, and taboo Lévy flights.

The fractional Laplacian (-Δ)^{α/2}, α∈(0,2), has many equivalent (albeit formally different) realizations as a nonlocal generator of a family of α-stable stochastic processes in R^{n}. On the other hand, if the process is to be restricted to a bounded domain, there are many inequivalent proposals for what a boundary-data-respecting fractional Laplacian should actually be. This ambiguity not only holds true for each specific choice of the process behavior at the boundary (e.

Publisher's Note: Killing (absorption) versus survival in random motion [Phys. Rev. E 96, 032104 (2017)].

This corrects the article DOI: 10.1103/PhysRevE.96.

Killing (absorption) versus survival in random motion.

We address diffusion processes in a bounded domain, while focusing on somewhat unexplored affinities between the presence of absorbing and/or inaccessible boundaries. For the Brownian motion (Lévy-stable cases are briefly mentioned) model-independent features are established of the dynamical law that underlies the short-time behavior of these random paths, whose overall lifetime is predefined to be long. As a by-product, the limiting regime of a permanent trapping in a domain is obtained.

Lévy flights in an infinite potential well as a hypersingular Fredholm problem.

We study Lévy flights with arbitrary index 0<μ≤2 inside a potential well of infinite depth. Such a problem appears in many physical systems ranging from stochastic interfaces to fracture dynamics and multifractality in disordered quantum systems. The major technical tool is a transformation of the eigenvalue problem for initial fractional Schrödinger equation into that for Fredholm integral equation with hypersingular kernel.