Excursions and Itôcalculus in Nelson’s Stochastic Mechanics

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


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,…,
$$P\left( {\# ^ \pm {\text{(s,t)}} = {\text{N}}} \right) = {\text{e}}^{{\text{ - t du}}^ \pm {\text{(s)}}} \frac{{(td\upsilon ^ \pm (s))^N }} {{N!}},$$
$$frac{{d{\upsilon ^ \pm }\left( s \right)}}{{ds}} = f_E^{ - 2}\left( a \right){\left( {{f_E},{{\left( {{H_ \pm } - E} \right)}^2}exp\left( { - s\left( {{H_ \pm } - E} \right)} \right){f_E}} \right)_L}2 $$
H± being a Dirichlet Hamiltonian for the Coulomb-type potential, with Dirichlet boundary conditions outside inside the sphere of radius a.


Stochastic Integral Stochastic Mechanics Stochastic Acceleration Continuous Sample Path Forward Kolmogorov Equation 
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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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