A Feynman-Kac-type formula for the deterministic and stochastic wave equations and other p.d.e.’s
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- by Robert C. Dalang, Carl Mueller and Roger Tribe PDF
- Trans. Amer. Math. Soc. 360 (2008), 4681-4703 Request permission
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.References
- Sergio A. Albeverio and Raphael J. Høegh-Krohn, Mathematical theory of Feynman path integrals, Lecture Notes in Mathematics, Vol. 523, Springer-Verlag, Berlin-New York, 1976. MR 0495901
- S. Albeverio, Ph. Blanchard, Ph. Combe, R. Høegh-Krohn, and M. Sirugue, Local relativistic invariant flows for quantum fields, Comm. Math. Phys. 90 (1983), no. 3, 329–351. MR 719294
- René A. Carmona and S. A. Molchanov, Parabolic Anderson problem and intermittency, Mem. Amer. Math. Soc. 108 (1994), no. 518, viii+125. MR 1185878, DOI 10.1090/memo/0518
- K. L. Chung and R. J. Williams, Introduction to stochastic integration, 2nd ed., Probability and its Applications, Birkhäuser Boston, Inc., Boston, MA, 1990. MR 1102676, DOI 10.1007/978-1-4612-4480-6
- Giuseppe Da Prato and Jerzy Zabczyk, Stochastic equations in infinite dimensions, Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, 1992. MR 1207136, DOI 10.1017/CBO9780511666223
- Robert C. Dalang, Extending the martingale measure stochastic integral with applications to spatially homogeneous s.p.d.e.’s, Electron. J. Probab. 4 (1999), no. 6, 29. MR 1684157, DOI 10.1214/EJP.v4-43
- Robert C. Dalang, 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. (electronic); MR1684157 (2000b:60132)], Electron. J. Probab. 6 (2001), no. 6, 5. MR 1825714
- Robert C. Dalang and N. E. Frangos, The stochastic wave equation in two spatial dimensions, Ann. Probab. 26 (1998), no. 1, 187–212. MR 1617046, DOI 10.1214/aop/1022855416
- Robert C. Dalang and Olivier Lévêque, Second-order hyperbolic S.P.D.E.’s driven by homogeneous Gaussian noise on a hyperplane, Trans. Amer. Math. Soc. 358 (2006), no. 5, 2123–2159. MR 2197451, DOI 10.1090/S0002-9947-05-03740-2
- Dalang, R.C. & Mueller, C. Intermittency properties in a hyperbolic Anderson problem (in preparation).
- Nicolas Fournier and Sylvie Méléard, A stochastic particle numerical method for 3D Boltzmann equations without cutoff, Math. Comp. 71 (2002), no. 238, 583–604. MR 1885616, DOI 10.1090/S0025-5718-01-01339-4
- Reuben Hersh, Random evolutions: a survey of results and problems, Rocky Mountain J. Math. 4 (1974), 443–477. MR 394877, DOI 10.1216/RMJ-1974-4-3-443
- Kac, M. Some stochastic problems in physics and mathematics. Magnolia Petroleum Co. Lectures in Pure and Applied Science 2 (1956).
- Mark Kac, A stochastic model related to the telegrapher’s equation, Rocky Mountain J. Math. 4 (1974), 497–509. Reprinting of an article published in 1956. MR 510166, DOI 10.1216/RMJ-1974-4-3-497
- Ioannis Karatzas and Steven E. Shreve, Brownian motion and stochastic calculus, 2nd ed., Graduate Texts in Mathematics, vol. 113, Springer-Verlag, New York, 1991. MR 1121940, DOI 10.1007/978-1-4612-0949-2
- Lévèque, O. Hyperbolic SPDE’s driven by boundary noises, Ph.D. Thesis No. 2452, Ecole Polytechnique Fédérale de Lausanne, Switzerland, 2001.
- Carl Mueller, Long time existence for the wave equation with a noise term, Ann. Probab. 25 (1997), no. 1, 133–151. MR 1428503, DOI 10.1214/aop/1024404282
- B. Øksendal, G. Våge, and H. Z. Zhao, Asymptotic properties of the solutions to stochastic KPP equations, Proc. Roy. Soc. Edinburgh Sect. A 130 (2000), no. 6, 1363–1381. MR 1809108, DOI 10.1017/S030821050000072X
- Mark A. Pinsky, Lectures on random evolution, World Scientific Publishing Co., Inc., River Edge, NJ, 1991. MR 1143780, DOI 10.1142/1328
- Laurent Schwartz, Théorie des distributions, Publications de l’Institut de Mathématique de l’Université de Strasbourg, IX-X, Hermann, Paris, 1966 (French). Nouvelle édition, entiérement corrigée, refondue et augmentée. MR 0209834
- John B. Walsh, An introduction to stochastic partial differential equations, École d’été de probabilités de Saint-Flour, XIV—1984, Lecture Notes in Math., vol. 1180, Springer, Berlin, 1986, pp. 265–439. MR 876085, DOI 10.1007/BFb0074920
- Richard L. Wheeden and Antoni Zygmund, Measure and integral, Pure and Applied Mathematics, Vol. 43, Marcel Dekker, Inc., New York-Basel, 1977. An introduction to real analysis. MR 0492146
- Robert L. Wolpert, Local time and a particle picture for Euclidean field theory, J. Functional Analysis 30 (1978), no. 3, 341–357. MR 518340, DOI 10.1016/0022-1236(78)90062-9
Additional Information
- 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
- Received by editor(s): October 13, 2005
- Received by editor(s) in revised form: May 19, 2006
- Published electronically: 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 2008 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 360 (2008), 4681-4703
- MSC (2000): Primary 60H15; Secondary 60H20
- DOI: https://doi.org/10.1090/S0002-9947-08-04351-1
- MathSciNet review: 2403701