Abstract
We study the asymptotic behavior of random time changes of dynamical systems. As random time changes we propose three classes which exhibits different patterns of asymptotic decays. The subordination principle may be applied to study the asymptotic behavior of the random time dynamical systems. It turns out that for the special case of stable subordinators explicit expressions for the subordination are known and its asymptotic behavior are derived. For more general classes of random time changes explicit calculations are essentially more complicated and we reduce our study to the asymptotic behavior of the corresponding Cesaro limit.
Highlights
In this paper we will deal with Markov processes or dynamical systems in Rd
For a dynamical system we introduce u(t, x ) = f ( X x (t))
Where Gt (τ ) is the density of the inverse subordinator E(t), see, e.g., Toaldo [2], Kondratiev and da Silva [1] and especially the book Meerschaert and Sikorskii [3]. This formula which relates the solutions of the evolution equations with usual and fractional derivatives plays an important role in the study of dynamics with random times
Summary
In this paper we will deal with Markov processes or dynamical systems in Rd. These processes or dynamics starting from x ∈ Rd , denote by X x (t), t ≥ 0, have associated evolution equations on Rd. Where Gt (τ ) is the density of the inverse subordinator E(t), see, e.g., Toaldo [2], Kondratiev and da Silva [1] and especially the book Meerschaert and Sikorskii [3] This formula which relates the solutions of the evolution equations with usual and fractional derivatives plays an important role in the study of dynamics with random times. For particular classes of random times the subordination formula (1) is evaluated explicitly This is true, for example, in the case of inverse stable subordinators. We would like to emphasize that for particular classes of random times, namely inverse stable subordinators, the asymptotic of u E (t, x ) which may be computed explicitly, coincides with the Cesaro limit.
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