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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 $$du(t) + Au(t)dt = \sum B^j u(t) dw^j(t) + f(t), \quad u(0) = u_0,$$ where $- A$ is a generator of an analytic semigroup $\{e^{- tA}\}_{r \ge 0}$ on $X$, an $M$-type 2 Banach space, $B^1, \ldots, B^d$ 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.

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
Full Text: DOI
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