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Zbl 1031.60061
Keck, David N.; McKibben, Mark A.
Functional integro-differential stochastic evolution equations in Hilbert space.
(English)
[J] J. Appl. Math. Stochastic Anal. 16, No.2, 141-161 (2003). ISSN 1048-9533; ISSN 1687-2177/e

Let $K$ and $H$ be real separable Hilbert spaces. Assume that $W$ is a $K$-valued Wiener process with covariance operator $Q$ and $x_0$ is an $H$-valued random variable which is independent of $W$. Consider the initial value problem of semilinear functional integro-differential stochastic evolution equations $$x^\prime (t) = A x(t) + F(x)(t) + \int^t_0 G(x)(s) dW(s),\ 0 \le t \le T, \quad x(0) = h(x) + x_0$$ with values in $H$, where $A : H \to H$ represents a linear operator, $G : C([0,T],H) \to C([0,T], L^2(\Omega,BL(K,H)))$, $F : C([0,T],H) \to L^p([0,T],L^2(\Omega,H))$ with $1\le p < + \infty$ and $h: C([0,T],H) \to L^2_0(\Omega,H)$. The authors discuss global existence results concerning mild and periodic solutions under several growth and compactness conditions. Weak convergence of induced probability measures belonging to the family of finite-dimensional distributions of certain sequences of such stochastic equations is treated too. Basic proof-tools include Schaefer's fixed point theorem, techniques of linear semigroups and probability measures as well as results from infinite-dimensional SDEs. Conceivable applications to electromagnetic theory, population dynamics and heat conduction in materials with memory underline the importance of their work. An example of a nonlocal integro-partial SDE illustrates some thoughts of related abstract theory. Some necessary preliminaries compiled from probability theory and functional analysis ease the process of understanding by lesser experienced readership.
[Henri Schurz (Carbondale)]
MSC 2000:
*60H25 Random operators, etc.
34F05 ODE with randomness
37H10 Random and stochastic difference and differential equations
37L55 Infinite-dimensional random dynamical systems etc.
60B05 Probability measures on topological spaces
60H15 Stochastic partial differential equations
60H20 Stochastic integral equations
60H30 Appl. of stochastic analysis
60H10 Stochastic ordinary differential equations
34K30 Functional-differential equations in abstract spaces

Keywords: semilinear stochastic evolution equations; mild solutions; semigroup techniques; Hilbert-space valued Wiener process; weak convergence of probability measures; stochastic equations with memory; Schaefer's fixed point theorem

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