Ruelle's Inequality for Isotropic Ornstein–uhlenbeck Flows

Abstract

Ruelle's inequality asserts that the entropy of a dynamical system is bounded from above by the Lyapunov characteristic numbers counted with their multiplicities. We show that this inequality holds true in the case of a random dynamical system deduced from an isotropic Ornstein-Uhlenbeck-flow (IOUF).

Pesin’s Formula for Random Dynamical Systems on $$\mathbf {R}^d$$ R d

“…This proof was extended in [19] to isotropic OrnsteinUhlenbeck flows, which can be seen as some special random dynamical system on R d . This proof can be extended to our more general situation assuming Assumption 5.…”

“…For details concerning the definition of {ξ xi } i∈N and Ω k,l see [19]. Then for any i ∈ N and x ∈ ξ xi we have…”

“…Let us remark that we can use the one-step discretization without loss of generality for our purposes. If we denote ν t := P • ϕ −1 0,t then [19,Corollary 3.3] implies that for every t ≥ 0 the entropy satisfies h µ (X + (R d , ν t )) = th µ (X + (R d , ν)). …”

“…First, we will bound the entropy from below following the proof of [13,Chapter IV] and using the results from the previous sections (see Section 8.1). The estimate from above (see Section 8.2) was established in [19] for certain stochastic flows, but its proof can be applied to our situation by changing only two estimates in the proof.…”

Asymptotic Growth of Spatial Derivatives of Isotropic Flows

“…We will keep the convention of speaking of an IOUF only if its drift c is not equal to zero. (see [2] or [3] for a discusion of this issue).…”