Markov processes and generalized Schroedinger equations

TL;DR

This paper develops a framework linking Markov processes with generalized Schrödinger equations, extending Nelson's stochastic mechanics to include Levy processes and providing new insights into their mathematical structure.

Contribution

It introduces a method to derive Hamiltonians of generalized Schrödinger equations from Markov process generators using Doob transformations and Dirichlet forms, broadening stochastic mechanics.

Findings

Established a connection between stationary wave functions and Markov processes.

Extended Nelson's stochastic mechanics to Levy-type processes.

Provided examples involving Cauchy noise and Levy-Schroedinger equations.

Abstract

Starting from the forward and backward infinitesimal generators of bilateral, time-homogeneous Markov processes, the self-adjoint Hamiltonians of the generalized Schroedinger equations are first introduced by means of suitable Doob transformations. Then, by broadening with the aid of the Dirichlet forms the results of the Nelson stochastic mechanics, we prove that it is possible to associate bilateral, and time-homogeneous Markov processes to the wave functions stationary solutions of our generalized Schroedinger equations. Particular attention is then paid to the special case of the Levy-Schroedinger equations and to their associated Levy-type Markov processes, and to a few examples of Cauchy background noise.