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Stochastic partial differential equations in M-type 2 Banach spaces. (English) Zbl 0831.35161

The author proves an existence and uniqueness theorem for a stochastic differential equation in M-type 2 Banach spaces of the following type

du(t)+Au(t)dt=B j u(t)dw j (t)+f(t),u(0)=u 0 ,

where -A is a generator of an analytic semigroup {e -tA } r0 on X, an M-type 2 Banach space, B 1 ,...,B d are linear operators in X and w(t) is a d-dimensional Wiener process. The author considers the case, when the space of initial conditions is some real interpolation space between the domain of A and X, and the case, when the space of initial conditions is X. Also the author considers the case, when X is a Hilbert space and applies the obtained results to stochastic parabolic equations.


MSC:
35R60PDEs with randomness, stochastic PDE
60H15Stochastic partial differential equations
46E35Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
47F05Partial differential operators
47D06One-parameter semigroups and linear evolution equations
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