Abstract
Linear skew-product semidynamical systems generated by random systems of delay differential equations are considered, both on a space of continuous functions as~well as on a space of $p$-summable functions. The main result states that in both cases, the Lyapunov exponents are identical, and that the Oseledets decompositions are related by natural embeddings.
Highlights
The theory of linear random skew-product semidynamical systems has become a powerful tool in the investigation of random linear parabolic PDEs of second order driven by a measurable dynamical system on a probability space
When the solution operator is compact assuming the summability of the coefficients of the PDE, we have an Oseledets decomposition: the separable Banach space decomposes into a countable direct sum of invariant measurable families of finite-dimensional vector subspaces which can be characterized as corresponding to solutions defined on the whole real line having given logarithmic growth rates (Lyapunov exponents) both in the future and in the past
When we consider systems of linear random delay differential equations z (t) = A(θtω) z(t) + B(θtω) z(t − 1), any “natural” space on which we define a linear skew-product semidynamical system must contain a space consisting of functions defined on [−1, 0] and taking values on RN
Summary
The theory of linear random skew-product semidynamical systems has become a powerful tool in the investigation of random linear parabolic PDEs of second order driven by a measurable dynamical system on a probability space. Mierczynski and Shen [13] and Mierczynski et al [15] prove, under adequate dynamical assumptions, the existence of a principal Floquet subspace and a generalized exponential separation decomposition when the fiber is a separable Banach space with separable dual This has to do with the dual skew-product semidynamical systems. The adjoint equation has the same properties as the original equation, and in many cases one needs only to prove “one half” of a theorem (for example, the existence of an Oseledets filtration, whereas the other half can be given by applying the theorem to the skew-product system generated by the adjoint equation; for a similar approach see Section 3 in Mierczynski and Shen [14]) This is not the case for delay differential equations. The importance of these results is that the geometrical methods of construction for the Oseledets subspaces, obtained in [10] for reflexive separable Banach spaces, as well as the estimates of Lyapunov exponents, can be applied on RN × Lp([−1, 0], RN ) and translated to C([−1, 0], RN ) by embedding
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
