On SPDE’s and superdiffusions. (English) Zbl 0904.60047

Krylov, N. V.

Ann. Probab. 25, No. 4, 1789-1809 (1997).

The author shows that superdiffusions, that is stochastic processes valued in the space of all finite measures on \((R^d,B(R^d))\), could be considered as solutions to Itô type stochastic partial differential equations which allows to include superdiffusions in the framework of classical stochastic analysis. For this purpose there are introduced special sets of frame functions \(\varphi_i(\mu,x), i\geq 1\), on \(L_2(R^d,\mu)\) and an additional probability space \((\widetilde\Omega,\widetilde{ F },\widetilde P)\) carrying independent one-dimensional Wiener processes \(\widetilde w^k_t, k\geq 1\). Then it is proved that if \(\mu_t\) is a super-Brownian process on a probability space \((\Omega,{ F},P)\), then on \((\Omega,{ F},P)\times(\widetilde\Omega,\widetilde{ F },\widetilde P)\) there exist independent Wiener processes \(w^k_t,k\geq 1\), such that \[ d\mu_t=\Delta \mu_tdt+\sum_{i=1}^{\infty}\varphi_i(\mu_t,\cdot)\mu_tdw^i_t. \] More general SPDEs for super-Brownian motions are considered as well. Many open questions arise from the approach suggested in the paper. Some of them are listed by the author.

Reviewer: Ya.Belopolskaya (St.Peterburg)

MSC:

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

Keywords:

superdiffusions; stochastic partial differential equations

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