Symmetric Ornstein-Uhlenbeck semigroups and their generators

Abstract

We provide necessary and sufficient conditions for a Hilbert space-valued Ornstein-Uhlenbeck process to be reversible with respect to its invariant measure µ. For a reversible process the domain of its generator in L p (µ) is characterized in terms of appropriate Sobolev spaces thus extending the Meyer equivalence of norms to any symmetric Ornstein-Uhlenbeck operator. We provide also a formula for the size of the spectral gap of the generator. Those results are applied to study the Ornstein-Uhlenbeck process in a chaotic environment. Necessary and sufficient conditions for a transition semigroup (R t ) to be compact, Hilbert-Schmidt and strong Feller are given in terms of the coefficients of the Ornstein-Uhlenbeck operator. We show also that the existence of spectral gap implies a smoothing property of R t and provide an estimate for the (appropriately defined) gradient of R t φ. Finally, in the Hilbert-Schmidt case, we show that for any φ ∈ L p (µ) the function R t φ is an (almost) classical solution of a version of the Kolmogorov equation.

Asymptotics of the Entropy Production Rate for d-Dimensional Ornstein–Uhlenbeck Processes

“…It is well known that the Markov process ξ has an invariant probability µ if and only if (2.3) all the eigenvalues of B have negative real parts, see [2,14], and in that case µ = N(0, Γ), (2.4) the Gaussian distribution on R d with mean vector 0 and covariance matrix Γ = ∞ 0 e sB Qe sB T ds. (2.5) Notice that Γ is nonsingular under conditions (2.2) and (2.3).…”

“…It is shown that the stationary OU process ξ is reversible if and only if the coefficients B and Q satisfy the symmetry condition Q −1 B = (Q −1 B) T , see [2,14]. Notice that when d = 1, B < 0, the OU process ξ is always reversible.…”

“…Thus, ξ is stationary. Let P [s,t] and P − [s,t] be the distributions of {ξ u } s≤u≤t and {ξ t+s−u } s≤u≤t for any 0 ≤ s ≤ t. Recall that the process ξ = {ξ t } t≥0 is said to be reversible if P [s,t] = P − [s,t] for any 0 ≤ s ≤ t. It is shown that the stationary OU process ξ is reversible if and only if the coefficients B and Q satisfy the symmetry condition [2,14]. Notice that when d = 1, B < 0, the OU process ξ is always reversible.…”

The Green–Kubo formula for general Markov processes with a continuous time parameter

“…In fact, L 2 -convergence, ergodicity and irreversibility with respect to the invariant measure for general Markov processes are sophisticated topics in probability theory (see [21] and references therein), which is not the main purpose of this paper. To illustrate the Green-Kubo formula, we present two typical classes of stochastic evolution equations in infinite dimensional spaces, of which L 2 convergence and irreversibility have been shown by Da Prato et al [6], Chojnowska-Michalik et al [3], Deuschel [11] and Feng et al [16]. One is a class of stochastic evolution equations in a separable Hilbert space, which has a unique invariant measure.…”

“…A is symmetric if and only if F = 1 2 ∇ log ρ, where ρ = dμ dν , the Radon-Nikodym derivative of μ with respect to ν, ν = N(0, Q ∞ ). (II) (Theorem 2.4 in [3]). Assume that F = 0.…”