Hostname: page-component-7dd5485656-dk7s8 Total loading time: 0 Render date: 2025-10-30T18:05:38.891Z Has data issue: false hasContentIssue false
  • English
  • Français

Moments of certain stochastic integrals occurring in mathematical physics

Published online by Cambridge University Press:  14 November 2011

Lieven Smits
Lieven Smits
Affiliation:
Departement Wiskunde, Universiteit Antwerpen (UIA), Universiteitsplein 1, 2610 Wilrijk, Belgium
Get access Rights & Permissions [Opens in a new window]

Synopsis

We give an expression for the n-th moment of certain Itô integrals. The integrands considered are nonanticipating functionals of the form s↦a(s, Xs), where a is a measurable time-dependent vector field in space satisfying mild regularity conditions, and Xs is standard translated Brownian motion. The expressions are similar to the Dyson-Phillips terms for magnetic Schrödinger semigroups.

We use these expressions to establish properties of the solutions of certain Cauchy problems and we relate our results to the framework of generalised Dyson expansions as set up by Johnson and Lapidus.

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 120 , Issue 3-4 , 1992 , pp. 267 - 282
Copyright
Copyright © Royal Society of Edinburgh 1992

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

1Aizenman, M. and Simon, B.. Brownian Motion and Harnack Inequality for Schrödinger Operators. Comm. Pure Appl. Math. 35 (1982), 209–273.CrossRefGoogle Scholar
2Albeverio, S. and Ma, Z. Additive functionals, nowhere Radon and Kato class smooth measures associated wtih Dirichlet forms (Preprint nr. 66, Sonderforschungsbereich 237, Institut für Mathematik Ruhr-Universität Bochum, Duitsland, October 1989).Google Scholar
3Dellacherie, C. and Meyer, P.-A.. Probabilités et potentiel (Paris: Hermann, 1980).Google Scholar
4Durrett, R.. Brownian Motion and Martingales in Analysis (Belmont: Wadsworth, 1984).Google Scholar
5Fukushima, M.. Dirichlet Forms and Markov Processes, North-Holland Mathematical Library 23 (Amsterdam: North-Holland, 1980).Google Scholar
6Hille, E. and Phillips, R. S.. Functional Analysis and Semi-groups, American Mathematical Society Colloqiuim Publications 31 (Providence R.I.: American Mathematical Society, 1957).Google Scholar
7John, F. and Nirenberg, L.. On functions of bounded mean oscillation. Comm. Pure Appl. Math. 14 (1961), 415–426.CrossRefGoogle Scholar
8Johnson, G. W. and Lapidus, M. L.. Generalized dyson series, generalized Feynman diagrams, the Feynman integral and Feynman's operational calculus. Mem. Amer. Math. Soc. 62, No. 351 (July 1986).Google Scholar
9Johnson, G. W. and Lapidus, M. L.. Noncommutative Operations on Wiener Functionals and Feynman's Operational Calculus. J. Fund. Anal. 81 (1988), 74–99.CrossRefGoogle Scholar
10Kac, M.. On some connections between probability theory and differential equations, Proc. 2nd Berk. Symp. Math. Statist. Probability (1950), 189–215.Google Scholar
11Lapidus, M. L.. The Feynman-Kac formula with a Lebesgue-Stieltjes measure and Feynman's operational calculus. Stud. Appl. Math. 76 (1987), 93–132.CrossRefGoogle Scholar
12McKean, H. P. Jr.Stochastic Integrals (New York: Academic Press, 1969).Google Scholar
13Schechter, M.. Spectral of Partial Differential Operators (Amsterdam: North-Holland, 1986).Google Scholar
14Simon, B.. Schrödinger Semigroups. Bull. Amer. Math. Soc. 7 (1982), 447–526.CrossRefGoogle Scholar
15Simon, B.. Functional Integration and Quantum Physics (New York: Academic Press, 1979).Google Scholar
16Smits, L.. A Remark on Smoothing of Magnetic Schrödinger Semigroups. Comm. Math. Phys. 123 (1989), 527–528.CrossRefGoogle Scholar
17Smits, L. and van Casteren, J. A.. Semigroups Defined by Additive Processes. UIA-preprint 90–07, march 1990. Proceedings of the 2nd International Conference on Trends in Semigroup Theory and Evolution Equations, Delft, The Netherlands, 25/9–29/9/1989 (New York: Marcel Dekker, to appear).Google Scholar
18van Casteren, J. A.. Generators of Strongly Continuous Semigroups. (Boston: Pitman, 1985).Google Scholar