×

Diffusion equations in duals of nuclear spaces. (English) Zbl 0702.60056

The authors study Ito type stochastic differential equations of processes with values in the dual \(E'\) of a nuclear Fréchet space E. Let \[ A: {\mathbb{R}}_+\times E'\to E'\text{ and } B: {\mathbb{R}}_+\times E'\to L(E',E') \] be given coefficient functions and let \((W_ t)_{t\geq 0}\) be an \(E'\)-valued Gaussian process with covariance operator Q. Under familiar conditions on the functions A and B it is proved that the stochastic differential equation \[ dX_ t=A(t,X_ t)dt+B(t,X_ t)dW_ t \] has unique solutions. The existence of the solutions is proved by studying the associated martingale problem and assuming certain growth conditions on A and B. It is shown that the family of solutions of associated finite-dimensional martingale problems is tight and that any limit point is a solution of the given martingale problem. Uniqueness is proved under Lipschitz conditions on A and B using the result of T. Yamada and S. Watanabe [J. Math. Kyoto Univ. 11, 155-167 (1971; Zbl 0236.60037); ibid., 553-563 (1971; Zbl 0229.60039)] that pathwise uniqueness implies uniqueness of the martingale problem.
An application to the motion of random strings is discussed as an example that duals of nuclear spaces are natural state spaces for many problems which can be described by stochastic differential equations.
Reviewer: E.Dettweiler

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60J60 Diffusion processes
60B05 Probability measures on topological spaces
60H20 Stochastic integral equations
60B11 Probability theory on linear topological spaces
PDFBibTeX XMLCite
Full Text: DOI