×

Uniqueness for a class of one-dimensional stochastic PDEs using moment duality. (English) Zbl 1044.60048

Summary: We establish a duality relation for the moments of bounded solutions to a class of one-dimensional parabolic stochastic partial differential equations. The equations are driven by multiplicative space-time white noise, with a non-Lipschitz multiplicative functional. The dual process is a system of branching Brownian particles. The same method can be applied to show uniqueness in law for a class of non-Lipschitz finite-dimensional stochastic differential equations.

MSC:

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)

Keywords:

duality; uniqueness
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] Athreya, S. (1998). Probability and semilinear partial differential equations. Ph.D. dissertation, Univ. Washington.
[2] Bass, R. (1995). Probabilistic Techniques in Analysis. Springer, New York. · Zbl 0817.60001
[3] Dawson, D. (1991). Measure valued Markov processes. Ecole \'d Eté de Probabilitiés de SaintFlour XXI. Lecture Notes in Math. 1541. Springer, New York.
[4] Ethier, S. and Kurtz, T. (1986). Markov Processes: Characterization and Convergence. Wiley, New York. · Zbl 0592.60049
[5] Griffeath, D. (1979). Additive and cancellative interacting particle systems. Lecture Notes in Math. 724. Springer, Berlin. · Zbl 0412.60095
[6] Konno, N. and Shiga, T. (1988). Stochastic differential equations for some measure-valued diffusions. Probab. Theory Related Fields 79 201-225. · Zbl 0631.60058
[7] Mueller, C. and Perkins, E. (1992). The compact support property for solutions to the heat equation with noise. Probab. Theory Related Fields 93 325-358. · Zbl 0767.60054
[8] Mueller, C. and Tribe, R. (1997). Finite width for a random stationary interface. Electronic J. Probab. 2. · Zbl 0890.60056
[9] Mytnik, L. (1998). Weak uniqueness for the heat equation with noise. Ann. Probab. 26 968-984. · Zbl 0935.60045
[10] Pardoux, E. (1993). Stochastic partial differential equations, a review. Bull. Sci. Math. 117 29-47. · Zbl 0777.60054
[11] Perkins, E. (1995). On the martingale problem for interactive measure-valued diffusions. Mem. Amer. Math. Soc. 115. · Zbl 0823.60071
[12] Reimers, M. (1989). One-dimensional stochastic partial differential equations and the branching measure diffusion. Probab. Theory Related Fields 81 319-340. · Zbl 0651.60069
[13] Shiga, T. (1988). Stepping stone models in population genetics and population dynamics. Stochastic Processes Phys. Engng. Math. Appl. 42 345-355. · Zbl 0656.92006
[14] Shiga, T. (1993). Two contrasting properties of solutions for one-dimensional stochastic partial differential equations. Canad. J. Math 46 415-437. · Zbl 0801.60050
[15] Tribe, R. (1996). A traveling wave solution to the Kolmogorov equation with noise. Stochastics Stochastic Rep. 56 317-340. · Zbl 1002.60555
[16] Walsh, J. (1984). An introduction to stochastic partial differential equations. Ecole \'d Eté de Probabilitiés de Saint-Flour XIV. Lecture Notes in Math. 1180. Springer, New York. · Zbl 0608.60060
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.