Abstract
We discuss existence, uniqueness, and space–time Hölder regularity for solutions of the parabolic stochastic evolution equation { d U ( t ) = ( A U ( t ) + F ( t , U ( t ) ) ) d t + B ( t , U ( t ) ) d W H ( t ) , t ∈ [ 0 , T 0 ] , 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 ] × E → E and B : [ 0 , T ] × E → 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 λ ( [ 0 , T ] ; D ( ( − A ) θ ) ) ) provided λ ⩾ 0 and θ ⩾ 0 satisfy λ + θ < 1 2 . Various extensions are given and the results are applied to parabolic stochastic partial differential equations.
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