Annals of Probability

Time-Reversal in Hyperbolic S.P.D.E.'s

Robert C. Dalang and John B. Walsh

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This paper studies questions of changes of variables in a class of hyperbolic stochastic partial differential equations in two variables driven by white noise. Two types of changes of variables are considered: naive changes of variables which do not involve a change of filtration, which affect the equation much as though it were deterministic, and changes of variables that do involve a change of filtration, such as time-reversals. In particular, if the process in reversed coordinates does satisfy an s.p.d.e., then we show how this s.p.d.e. is related to the original one. Time-reversals for the Brownian sheet and for equations with constant coefficients are considered in detail. A necessary and sufficient condition is provided under which the reversal of the solution to the simplest hyperbolic s.p.d.e. with certain random initial conditions again satisfies such an s.p.d.e. This yields a negative conclusion concerning the reversal in time of the solution to the stochastic wave equation (in one spatial dimension) driven by white noise.

Article information

Ann. Probab., Volume 30, Number 1 (2002), 213-252.

First available in Project Euclid: 29 April 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H15: Stochastic partial differential equations [See also 35R60]
Secondary: 60G15: Gaussian processes 35R60: Partial differential equations with randomness, stochastic partial differential equations [See also 60H15]

Hyperbolic stochastic partial differential equations time reversal changes of variables Brownian sheet infinite dimensional diffusions


Dalang, Robert C.; Walsh, John B. Time-Reversal in Hyperbolic S.P.D.E.'s. Ann. Probab. 30 (2002), no. 1, 213--252. doi:10.1214/aop/1020107766.

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  • [1] BROCKHAUS, O. (1993). Sufficient statistics for the Brownian sheet. Séminaire de Probabilités XXVII. Lecture Notes in Math. 1557 44-52. Springer, Berlin.
  • [2] CAIROLI, R. and WALSH, J. B. (1975). Stochastic integrals in the plane. Acta Math. 134 111- 183.
  • [3] DALANG, R. C. and RUSSO, F. (1988). A prediction problem for the Brownian sheet. J. Multivariate Anal. 26 16-47.
  • [4] FARRÉ, M. and NUALART, D. (1993). Nonlinear stochastic integral equations in the plane. Stochastic Process. Appl. 46 219-239.
  • [5] FÖLLMER, H. and WAKOLBINGER, A. (1986). Time reversal of infinite-dimensional diffusions. Stochastic Process. Appl. 22 59-77.
  • [6] GOURSAT, E. (1964). A Course in Mathematical Analysis Vol. III. Dover, New York.
  • [7] HARDY, G. H. (1974). On the convergence of certain multiple series. In Collected Papers of G. H. Hardy Vol. VI 5-10. Clarendon Press, Oxford.
  • [8] HAUSSMAN, U. G. and PARDOUX, E. (1986). Time reversal of diffusions. Ann. Probab. 14 1188-1205.
  • [9] IMKELLER, P. (1988). Two-Parameter Martingales and Their Quadratic Variation. Lecture Notes in Math. 1308. Springer, Berlin.
  • [10] JEULIN, TH. and YOR, M. (1992). Une décomposition non-canonique du drap brownien. Séminaire de Probabilités XXVI. Lecture Notes in Math. 1526 322-347. Springer, Berlin.
  • [11] MILLET, A., NUALART, D. and SANZ, M. (1989). Time reversal for infinite-dimensional diffusions. Probab. Theory Related Fields 82 315-347.
  • [12] MOORE, CH. N. (1973). Summable Series and Convergence Factors. Dover, New York.
  • [13] REVUZ, D. and YOR, M. (1999). Continuous Martingales and Brownian Motion, 3rd ed. Springer, Berlin.
  • [14] ROVIRA, C. and SANZ-SOLÉ, M. (1995). A nonlinear hyperbolic SPDE: approximations and support. In Stochastic Partial Differential Equations (A. Etheridge, ed.) 241-261. Cambridge Univ. Press.
  • [15] ROVIRA, C. and SANZ-SOLÉ, M. (1996). The law of the solution to a nonlinear hyperbolic SPDE. J. Theoret. Probab. 9 863-901.
  • [16] WALSH, J. B. (1986). An Introduction to Stochastic Partial Differential Equations. École d'été de probabilités de Saint-Flour XIV. Lecture Notes in Math. 1180 265-439. Springer, Berlin.