Transactions of the American Mathematical Society

A Feynman-Kac-type formula for the deterministic and stochastic wave equations and other p.d.e.'s

Author(s): Robert C. Dalang; Carl Mueller; Roger Tribe
Journal: Trans. Amer. Math. Soc. 360 (2008), 4681-4703.
MSC (2000): Primary 60H15; Secondary 60H20
Posted: April 14, 2008
Retrieve article in: PDF DVI PostScript

Abstract: We establish a probabilistic representation for a wide class of linear deterministic p.d.e.'s with potential term, including the wave equation in spatial dimensions 1 to 3. Our representation applies to the heat equation, where it is related to the classical Feynman-Kac formula, as well as to the telegraph and beam equations. If the potential is a (random) spatially homogeneous Gaussian noise, then this formula leads to an expression for the moments of the solution.

1.: Albeverio, S.A. & Hoegh-Krohn, R.J. Mathematical theory of Feynman path integrals. Lecture Notes in Mathematics, Vol. 523. Springer-Verlag, Berlin-New York, 1976. MR 0495901 (58:14535)
2.: Albeverio, S., Blanchard, Ph., Combe, Ph., Hoegh-Krohn, R. & Sirugue, M. Local relativistic invariant flows for quantum fields. Comm. Math. Phys. 90 (1983), 329-351. MR 719294 (86i:81071)
3.: Carmona, R.A. & Molchanov, S.A. Parabolic Anderson problem and intermittency. Mem. Amer. Math. Soc. 108 (1994), no. 518. MR 1185878 (94h:35080)
4.: Chung, K.L. & Williams, R.J. Introduction to stochastic integration. Second edition. Probability and its Applications. Birkhäuser Boston, Inc., Boston, MA, 1990. MR 1102676 (92d:60057)
5.: Da Prato, G. & Zabczyk, J. Stochastic equations in infinite dimensions. Encyclopedia of Mathematics and its Applications, 44. Cambridge University Press, Cambridge, 1992. MR 1207136 (95g:60073)
6.: Dalang, R.C. Extending the martingale measure stochastic integral with applications to spatially homogeneous s.p.d.e.'s. Electron. J. Probab. 4 (1999), no. 6, 29 pp. MR 1684157 (2000b:60132)
7.: Dalang, R.C. Corrections to: ``Extending the martingale measure stochastic integral with applications to spatially homogeneous s.p.d.e.'s" Electron J. Probab. 4 (1999), no. 6, 29 pp. MR 1825714 (2002b:60111)
8.: Dalang, R.C. & Frangos, N.E. The stochastic wave equation in two spatial dimensions. Ann. Probab. 26 (1998), no. 1, 187-212. MR 1617046 (99c:60127)
9.: Dalang, R.C. & Lévèque, O. Second-order hyperbolic s.p.d.e.'s driven by homogeneous Gaussian noise on a hyperplane. Transactions of the American Mathematical Society 358 (2006), no. 5, 2123-2159. MR 2197451 (2006k:60112)
10.: Dalang, R.C. & Mueller, C. Intermittency properties in a hyperbolic Anderson problem (in preparation).
11.: Fournier, N. & Méléard, Sylvie. A stochastic particle numerical method for Boltzmann equations without cutoff. Math. Comp. 71 (2002), no. 238, 583-604. MR 1885616 (2003a:60128)
12.: Hersch, R. Random evolutions: a survey of results and problems. Rocky Mountain J. Math. 4 (1974), 443-477. MR 0394877 (52:15676)
13.: Kac, M. Some stochastic problems in physics and mathematics. Magnolia Petroleum Co. Lectures in Pure and Applied Science 2 (1956).
14.: Kac, M.A. A stochastic model related to the telegrapher's equation. Rocky Mountain J. Math. 4 (1974), 497-509. MR 0510166 (58:23185)
15.: Karatzas, I. & Shreve, S.E. Brownian motion and stochastic calculus. Second edition. Graduate Texts in Mathematics, 113. Springer-Verlag, New York, 1991. MR 1121940 (92h:60127)
16.: Lévèque, O. Hyperbolic SPDE's driven by boundary noises, Ph.D. Thesis No. 2452, Ecole Polytechnique Fédérale de Lausanne, Switzerland, 2001.
17.: Mueller, C. Long time existence for the wave equation with a noise term. Ann. Probab. 25 (1997), no. 1, 133-151. MR 1428503 (98b:60113)
18.: Oksendal, B., Vage, G. & Zhao, H.Z. Asymptotic properties of the solutions to stochastic KPP equations. Proc. Roy. Soc. Edinburgh Sect. A 130 (2000), no. 6, 1363-1381. MR 1809108 (2002g:60100)
19.: Pinsky, M.A. Lectures on random evolution. World Scientific, 1991. MR 1143780 (93b:60160)
20.: Schwartz, L. Théorie des distributions. Hermann, Paris, 1966. MR 0209834 (35:730)
21.: Walsh, J.B. An introduction to stochastic partial differential equations. Ecole d'été de probabilités de Saint-Flour, XIV--1984, Lecture Notes in Math., 1180, Springer, Berlin, 1986, 265-439. MR 876085 (88a:60114)
22.: Wheeden, R.L. & Zygmund, A. Measure and integral. An introduction to real analysis. Pure and Applied Mathematics, Vol. 43. Marcel Dekker, Inc., New York-Basel, 1977. MR 0492146 (58:11295)
23.: Wolpert, R.L. Local time and a particle picture for Euclidean field theory. J. Funct. Anal. 30-3 (1978), 341-357. MR 518340 (80a:81082)

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 60H15, 60H20

Robert C. Dalang
Affiliation: Institut de Mathématiques, Ecole Polytechnique Fédérale, Station 8, 1015 Lausanne, Switzerland
Email: robert.dalang@epfl.ch

Carl Mueller
Affiliation: Department of Mathematics, University of Rochester, Rochester, New York 14627
Email: cmlr@math.rochester.edu

Roger Tribe
Affiliation: Department of Mathematics, University of Warwick, CV4 7AL, United Kingdom
Email: tribe@maths.warwick.ac.uk

DOI: 10.1090/S0002-9947-08-04351-1
PII: S 0002-9947(08)04351-1
Keywords: Feynman-Kac formula, wave equation, probabilistic representation of solutions, stochastic partial differential equations, moment formulae.
Received by editor(s): October 13, 2005
Received by editor(s) in revised form: May 19, 2006
Posted: April 14, 2008
Additional Notes: The first author was partially supported by the Swiss National Foundation for Scientific Research
The second author was partially supported by an NSF grant.
Copyright of article: Copyright 2008, American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6850(e) ISSN 0002-9947(p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2008, American Mathematical Society Privacy Statement		Search the AMS