Partial smoothing of delay transition semigroups acting on special functions. (English) Zbl 1491.60087
Summary: It is well known that the transition semigroup of an Ornstein Uhlenbeck process with delay is not strong Feller for small times, so it has no regularizing effects when acting on bounded and continuous functions. In this paper we study regularizing properties of this transition semigroup when acting on special functions of the past trajectory. With this regularizing property, we are able to prove existence and uniqueness of a mild solution for a special class of semilinear Kolmogorov equations; we apply these results to a stochastic optimal control problem.
MSC:
| 60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |
| 34K50 | Stochastic functional-differential equations |
| 35R60 | PDEs with randomness, stochastic partial differential equations |
| 47D06 | One-parameter semigroups and linear evolution equations |
| 93E20 | Optimal stochastic control |
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