UNIFORMLY gamma-RADONIFYING FAMILIES OF OPERATORS AND THE STOCHASTIC WEISS CONJECTURE

作者：Haak Bernhard H^*; Van Neerven Jan

来源：Operators and Matrices, 2012, 6(4): 767-792.

DOI：10.7153/oam-06-50

• Uniformly gamma-radonifying families of operators
• R-boundedness
• Laplace transforms
• stochastic evolution equations
• invariant measures
• stochastic Weiss conjecture

We introduce the notion of uniform gamma-radonification of a family of operators, which unifies the notions of R-boundedness of a family of operators and gamma-radonification of an individual operator. We study the properties of uniformly gamma-radonifying families of operators in detail and apply our results to the stochastic abstract Cauchy problem %26lt;br%26gt;dU(t) = AU(t)dt + BdW(t), U(0) = 0. %26lt;br%26gt;Here, A is the generator of a strongly continuous semigroup of operators on a Banach space E, B is a bounded linear operator from a separable Hilbert space H into E, and W-H is an H-cylindrical Brownian motion. When A and B are simultaneously diagonalisable, we prove that an invariant measure exists if and only if the family %26lt;br%26gt;{root lambda R(lambda,A)B : lambda is an element of S-v} %26lt;br%26gt;is uniformly gamma-radonifying for some/all 0 %26lt; v %26lt; pi/2, where S-v is the open sector of angle v in the complex plane. This result can be viewed as a partial solution of a stochastic version of the Weiss conjecture in linear systems theory.

• 出版日期2012-12