Stochastic evolution equations in UMD Banach spaces
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We discuss existence, uniqueness, and space-time Hölder regularity for solutions of the parabolic stochastic evolution equation dU(t) = (AU(t) + F(t,U(t))) dt + B(t,U(t)) dW_H(t), t\in [0,\Tend], U(0) = u_0, where $A$ generates an analytic $C_0$-semigroup on a UMD Banach space $E$ and $W_H$ is a cylindrical Brownian motion with values in a Hilbert space $H$. We prove that if the mappings $F:[0,T]\times E\to E$ and $B:[0,T]\times E\to \mathscr{L}(H,E)$ satisfy suitable Lipschitz conditions and $u_0$ is $\F_0$-measurable and bounded, then this problem has a unique mild solution, which has trajectories in $C^ł([0,T];\D((-A)^θ)$ provided $λ\ge 0$ and $θ\ge 0$ satisfy $ł+θ<\frac12$. Various extensions of this result are given and the results are applied to parabolic stochastic partial differential equations.
Accepted for publication in Journal of Functional Analysis
Accepted for publication in Journal of Functional Analysis