0000000000385627
AUTHOR
P. Garbaczewski
Fractional Laplacians and L\'{e}vy Flights in Bounded Domains
We address Lévy-stable stochastic processes in bounded domains, with a focus on a discrimination between inequivalent proposals for what a boundary data-respecting fractional Laplacian (and thence the induced random process) should actually be. Versions considered are: the restricted Dirichlet, spectral Dirichlet and regional (censored) fractional Laplacians. The affiliated random processes comprise: killed, reflected and conditioned Lévy flights, in particular those with an infinite life-time. The related concept of quasi-stationary distributions is briefly mentioned.
Nonequilibrium dynamics of nonconservative diffusion processes
Fokker-Planck operators of diffusion processes with nonconservative drift fields, in dimension $N\geq 2$, can be directly related with non-Hermitian electromagnetic-type Hamiltonian generators of motion. The induced nonequilibrium dynamics of probability densities points towards an issue of path integral solutions of the Fokker-Planck equation, and calls for revisiting links between known exact path integral formulas for quantum propagators in real and Euclidean time, with these for Fokker-Planck-induced transition probability density functions. In below we shall follow the $N=3$ "magnetic thread", within which one encounters formally and conceptually distinct implementations of the magneti…