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Mean-square random dynamical systems. (English) Zbl 1267.37018

Two-parameter semigroups operating on the space \(L^2(\Omega,P;\mathbb R^d)\) of square-integrable \(\mathbb R^d\)-valued random variables are introduced in order to investigate random or stochastic differential equations with non-local coefficients. Assuming exponential contraction of differences of trajectories in the course of time, uniform in the initial time and locally uniform in \(L^2\), existence of a limiting point process in \(L^2\) is shown. This can be interpreted as a ‘one point attractor’ for the two-parameter semigroup, both pullback and forward (see [A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for infinite-dimensional non-autonomous dynamical systems. Berlin: Springer (2013; Zbl 1263.37002)]). Two one-dimensional examples of stochastic differential equations are discussed. For general situations problems with this approach arise from the difficulty of characterising compactness in \(L^2\).

MSC:

37B55 Topological dynamics of nonautonomous systems
37H99 Random dynamical systems
37L99 Infinite-dimensional dissipative dynamical systems
60H30 Applications of stochastic analysis (to PDEs, etc.)

Citations:

Zbl 1263.37002
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