The talk addresses the problem of smooth linearization for random dynamical systems in Banach spaces. Such smooth conjugacy not only preserves the topological structure of the system but also enables control over its local behavior, including stability and sensitivity to perturbations.
Specifically, we consider dynamical systems generated by random semilinear Carath{\'e}odory differential equations and stochastic differential equations with a uniformly exponentially stable linear part, where the growth rates are not assumed to be constant. We present sufficient conditions for strong topological equivalence between the given systems and their linearizations. Additionally, we establish criteria for the conjugating map to be a $C^k$ diffeomorphism between the aforementioned systems. |