Stochastic Evolution Equations in Hilbert Spaces
This chapter deals with semilinear stochastic evolution equations in Hilbert spaces. The first two sections are more introductory and review some important properties of Wiener processes and the stochastic Itô integral in Hilbert spaces. The main referenc
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Stochastic Evolution Equations in Hilbert Spaces
This chapter deals with semilinear stochastic evolution equations in Hilbert spaces. The first two sections are more introductory and review some important properties of Wiener processes and the stochastic Itô integral in Hilbert spaces. The main references for the presented material are [18, 61]. In Sect. 2.3 we introduce the general form of the semilinear stochastic evolution equations from [18, Chap. 7.1] which we treat numerically in the following chapters. Besides some more explicit examples this section also contains our usual set of assumptions which we use in order to derive all results in this chapter. In Sect. 2.4 we present an existence and uniqueness result. Although our assumptions on the nonlinearities are slightly more general as in [41] the presented proof follows basically the same idea as the proof of [41, Th. 1]. But since we give a more detailed regularity analysis of the mild solution in the two subsequent sections, our proof is slightly simplified. In Sects. 2.5 and 2.6 we precisely determine the spatial and temporal regularity properties of the mild solution. These results first appeared in [50]. In the final section of this chapter, we also discuss some possible generalizations of our results.
2.1 Hilbert Space-Valued Wiener Processes This section gives a short review on Hilbert space-valued Wiener processes. All results are well-known in the literature, for example, in [18, Chap. 4.1] and [61, Chap. 2.1]. By .˝; F ; P/ we denote a probability space and by .U; .; /U ; k kU / a separable Hilbert space. We also consider a linear, bounded, self-adjoint, positive semidefinite operator Q 2 L.U /. The trace of Q is given by Tr.Q/ WD
X
.Qei ; ei /U ;
(2.1)
i 2N
R. Kruse, Strong and Weak Approximation of Semilinear Stochastic Evolution Equations, Lecture Notes in Mathematics 2093, DOI 10.1007/978-3-319-02231-4__2, © Springer International Publishing Switzerland 2014
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2 Stochastic Evolution Equations in Hilbert Spaces
where .ei /i 2N is an arbitrary orthonormal basis of U . If the series Tr.Q/ is (absolutely) convergent, the operator Q is called trace class and the real number Tr.Q/ < 1 does not depend on the particular choice of the orthonormal basis. Further, from [18, Prop. C.3] it follows that every self-adjoint, positive semidefinite operator Q 2 L.U / with finite trace is compact and, therefore, the spectral theorem for compact operators [69, Th. VI.3.2] yields the existence of an orthonormal basis .ei /i 2N of U and a decreasing sequence of nonnegative real numbers .i /i 2N with i ! 0 as i ! 1 such that Qei D i ei ;
for all i 2 N;
and Qu D
X
i .u; ei /U ei ;
for all u 2 U:
i 2N
In particular, it holds Tr.Q/ D
X
i :
i 2N
We now define a (standard) Q-Wiener process in the same way as in [18, Chap. 4.1] and [61, Def. 2.1.9]. Definition 2.1. Let T > 0. A stochastic process W W Œ0; T ˝ ! U on .˝; F ; P/ is called a (standard) Q-Wiener process if (i) W .0/ D 0, (ii) W has P-a.s. continuous trajectories, (iii) W
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