Excursions and Itôcalculus in Nelson’s Stochastic Mechanics

Abstract

Using a simple-minded approach, we give a more or less self-contained account of Itô calculus and excursion theory in stochastic mechanics. We present some new results on Poisson-Lévy excursion measures for radial ground-state Nelson diffusions in Coulomb-type potentials: for these diffusions we consider excursions from a spherical shell of radius a. Let #^±(s,t) be the number of ^outward_inward excursions of duration s upto the local time at a equals t for our diffusion X corresponding to the radial ground-state wave-function f_E. Then, for N = 0,1,2,…,

$$P\left( {\# ^ \pm {\text{(s,t)}} = {\text{N}}} \right) = {\text{e}}^{{\text{ - t du}}^ \pm {\text{(s)}}} \frac{{(td\upsilon ^ \pm (s))^N }} {{N!}},$$

Where

$$frac{{d{\upsilon ^ \pm }\left( s \right)}}{{ds}} = f_E^{ - 2}\left( a \right){\left( {{f_E},{{\left( {{H_ \pm } - E} \right)}^2}exp\left( { - s\left( {{H_ \pm } - E} \right)} \right){f_E}} \right)_L}2 $$

H_± being a Dirichlet Hamiltonian for the Coulomb-type potential, with Dirichlet boundary conditions ^inside_outside the sphere of radius a.