Stochastic partial differential equations in infinite dimensional spaces.

*(English)*Zbl 0664.60062
Pisa: Scuola Normale Superiore. 144 p. (1988).

Martingale problems in infinite-dimensional spaces have been thoroughly investigated for years but there are few results in this direction for stochastic PDE. This book is the first exposition of the topic, much in the spirit of the pioneering thesis of M. Viot [Solutions faibles d’équations aux dérivées partielles stochastiques non linéaires. Paris 6 (1976)]. The heart of the book are chapters 6 and 7 where the stochastic differential equation \(dX=A(X)dt+B(X)dW\) is considered on some triple of Hilbert spaces \(V\subset H\subset V'\) with continuous and dense injections. It is assumed that W is a Wiener process on H with nuclear covariance while A and B are nonlinear, generally unbounded, operators satisfying certain mild continuity assumptions, moreover A is a coercive operator. It is proved that there exists a weak (in the sense of measure) solution of this equation provided one of the following conditions is satisfied:

1. The injection of V into H is compact and growth of A and B is at most linear.

2. A is a monotone operator with polynomial growth and B is of linear growth.

The idea of the proof is to construct approximating sequences of finite- dimensional pure jump Markov chains the laws of which form a relatively compact set in an appropriately chosen Skorokhod space. To this end an infinite-dimensional invariance principle is proved and some tightness criteria are formulated for Hilbert space-valued semimartingales. Existence of limit laws of the approximating sequences being established in this way, the next problem is to show that each limit law solves the martingale problem considered. This is achieved exactly as in Viot.

Chapters 1 to 5 contain some basic facts concerning infinite-dimensional martingale problems, construction of some useful canonical spaces and tightness of sequences of probability measures on some Lusin spaces.

1. The injection of V into H is compact and growth of A and B is at most linear.

2. A is a monotone operator with polynomial growth and B is of linear growth.

The idea of the proof is to construct approximating sequences of finite- dimensional pure jump Markov chains the laws of which form a relatively compact set in an appropriately chosen Skorokhod space. To this end an infinite-dimensional invariance principle is proved and some tightness criteria are formulated for Hilbert space-valued semimartingales. Existence of limit laws of the approximating sequences being established in this way, the next problem is to show that each limit law solves the martingale problem considered. This is achieved exactly as in Viot.

Chapters 1 to 5 contain some basic facts concerning infinite-dimensional martingale problems, construction of some useful canonical spaces and tightness of sequences of probability measures on some Lusin spaces.

Reviewer: B.Gołdys

##### MSC:

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

60B11 | Probability theory on linear topological spaces |