Markov processes and generalized 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.

33 pages, 2 figures; corrected as required by referee; references added; in press on J. Math. Phys

References in corpus (6)

• Fractional Quantum Mechanics
• Exact Mapping of the 2+1 Dirac Oscillator onto the Jaynes-Cummings Model: Ion-Trap Experimental Proposal
• Generalization of symmetric $\alpha$-stable Lévy distributions for $q>1$
• Selfdecomposability and selfsimilarity: a concise primer
• Mixtures in non stable Levy processes
• Lévy-Schrödinger wave packets