The stability of stochastic partial differential equations and applications. Theorems on supports.

*(English)*Zbl 0683.93092
Stochastic partial differential equations and applications II, Proc. 2nd Conf., Trento/Italy 1988, Lect. Notes Math. 1390, 91-118 (1989).

Summary: [For the entire collection see Zbl 0669.00018.]

We present a general result on the stability of stochastic evolution equations and of stochastic partial differential equations with respect to the simultaneous perturbations of the driving semimartingales and of the unbounded operators in the equations, in the topology of uniform convergence on finite time intervals in probability. Hence we obtain theorems on supports for stochastic evolution equations and stochastic partial differential equations. These results are generalizations of the Stroock-Varadhan support theorem of diffusion processes [see D. W. Stroock and S. R. S. Varadhan, Proc. 6th Berkeley Sympos. Math. Stat. Probab., Univ. Calif. 1970, 3, 333-359 (1972; Zbl 0255.60056)]. As applications, we prove theorems on supports for the nonlinear filter in the filtering theory of diffusion processes.

We present a general result on the stability of stochastic evolution equations and of stochastic partial differential equations with respect to the simultaneous perturbations of the driving semimartingales and of the unbounded operators in the equations, in the topology of uniform convergence on finite time intervals in probability. Hence we obtain theorems on supports for stochastic evolution equations and stochastic partial differential equations. These results are generalizations of the Stroock-Varadhan support theorem of diffusion processes [see D. W. Stroock and S. R. S. Varadhan, Proc. 6th Berkeley Sympos. Math. Stat. Probab., Univ. Calif. 1970, 3, 333-359 (1972; Zbl 0255.60056)]. As applications, we prove theorems on supports for the nonlinear filter in the filtering theory of diffusion processes.

##### MSC:

93E15 | Stochastic stability in control theory |

60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |

35R60 | PDEs with randomness, stochastic partial differential equations |

93E11 | Filtering in stochastic control theory |