In the present paper we consider the transition semigroup Pt related to some stochastic reaction-diffusion equations with the non-linear term f having polynomial growth and satisfying some dissipativity conditions. We are proving that it has a regularizing effect in the Banach space of continuous functions C(script O sign̄), where script O sign ⊂ double-struck R signd is a bounded open set. In L2(script O sign) the only result proved is the strong Feller property, for d = 1. Here we are able to prove that if f ∈ C∞(double-sruck R sign) and d ≤ 3, then Ptφ ∈ C∞b (C(script O sign̄)) for any φ ∈ Bb(C(script O sign̄)) and t > 0. An important application is to the study of the ergodic properties of the system. These results are also of interest for some problem in stochastic control.

Smoothing properties of transition semigroups relative to SDEs with value in Banach spaces / S. CERRAI. - In: PROBABILITY THEORY AND RELATED FIELDS. - ISSN 0178-8051. - STAMPA. - 113:(1999), pp. 85-114. [10.1007/s004400050203]

Smoothing properties of transition semigroups relative to SDEs with value in Banach spaces

CERRAI, SANDRA
1999

Abstract

In the present paper we consider the transition semigroup Pt related to some stochastic reaction-diffusion equations with the non-linear term f having polynomial growth and satisfying some dissipativity conditions. We are proving that it has a regularizing effect in the Banach space of continuous functions C(script O sign̄), where script O sign ⊂ double-struck R signd is a bounded open set. In L2(script O sign) the only result proved is the strong Feller property, for d = 1. Here we are able to prove that if f ∈ C∞(double-sruck R sign) and d ≤ 3, then Ptφ ∈ C∞b (C(script O sign̄)) for any φ ∈ Bb(C(script O sign̄)) and t > 0. An important application is to the study of the ergodic properties of the system. These results are also of interest for some problem in stochastic control.
1999
113
85
114
S. CERRAI
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Utilizza questo identificatore per citare o creare un link a questa risorsa: https://hdl.handle.net/2158/310261
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