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Excursions and Itôcalculus in Nelson’s Stochastic Mechanics

  • Aubrey Truman
  • David Williams
Part of the Mathematical Physics Studies book series (MPST, volume 12)

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 inward outward excursions of duration s upto the local time at a equals t for our diffusion X corresponding to the radial ground-state wave-function fE. 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 outside inside the sphere of radius a.

Keywords

Stochastic Integral Stochastic Mechanics Stochastic Acceleration Continuous Sample Path Forward Kolmogorov Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media Dordrecht 1991

Authors and Affiliations

  • Aubrey Truman
    • 1
  • David Williams
    • 2
  1. 1.Department of Mathematics and Computer ScienceUniversity College SwanseaSwanseaUK
  2. 2.Department of Pure Mathematics and Mathematical StatisticsUniversity of CambridgeCambridgeUK

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