Weak uniqueness for the heat equation with noise. (English) Zbl 0935.60045

Summary: The uniqueness in law for the equation \(\partial X_t/ \partial t={1\over 2}\Delta X_t+ X^\gamma_t\dot W\) is established for \(1/2<\gamma<1\). The proof uses a duality technique and requires the construction of an approximating sequence of dual processes.


60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35R60 PDEs with randomness, stochastic partial differential equations
Full Text: DOI


[1] Dawson, D. (1977). The critical measure diffusion process.Wahr. Verw Gebiete 40 125- 145. · Zbl 0343.60001
[2] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley, New York. · Zbl 0592.60049
[3] Fleischmann, K. (1988). Critical behavior of some measure-valued processes. Math. Nachr. 135 131-147. · Zbl 0655.60071
[4] Ikeda, N. and Watanabe, S. (1989). Stochastic Differential Equations and Diffusion Processes. North-Holland, Amsterdam. · Zbl 0684.60040
[5] Iscoe, I. (1986). A weighted occupation time for a class of measure-valued branching processes. Probab. Theory Related Fields 71 85-116. · Zbl 0555.60034
[6] Mueller, C. The heat equation with Ĺevy noise. Unpublished manuscript. · Zbl 0934.60056
[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] Mytnik, L. (1996). Superprocesses in random environments. Ann. Probab. 24 1953-1978. · Zbl 0874.60041
[9] Walsh, J. (1986). An Introduction to Stochastic Partial Differential Equations. Lecture Notes in Math. 1180 265-439. Springer, Berlin. · 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.