Asymptotic coupling and a general form of Harris’ theorem with applications to stochastic delay equations. (English) Zbl 1238.60082

Hairer, M.; Mattingly, J. C.; Scheutzow, M.

Probab. Theory Relat. Fields 149, No. 1-2, 223-259 (2011).

The paper under review studies the unique invariant measure and the existence of a spectral gap under conditions reminiscent of Harris’ theorem.
A Markov process \(\{X_t\}\) on the space \({\mathcal C}([-r, 0], {\mathbb R})\) possesses an invariant measure for a sufficiently large scale of drifting. For any solution \(X_t\) of a stochastic delay differential equation (SDDE) and for any \(t>0\), the initial condition \(X_0\) can be recovered with probability one. If the recovered initial condition in \({\mathcal C}([-r, 0], {\mathbb R})\) does not agree with \(X_0\), then the transition probabilities for any step of this chain are always mutually singular.
Theorem 3.1 in the reviewed paper provides sufficient conditions on the drift and the diffusion on the phase space to guarantee the existence of a unique global solution. The main idea is to use the results and techniques in [Weinan E, J. C. Mattingly and Y. Sinai, Commun. Math. Phys, 224, No. 1, 83–106 (2001; Zbl 0994.60065)]. E, Mattingly and Sinai reduce the original stochastic Navier-Stokes equation to an SDDE. The central idea is to “Girsanov-shift” the driving Wiener process to start at a different initial condition asymptotically as time goes to infinity.
In Section 2, the authors prove a very general theorem (Theorem 1.1) which gives a verifiable condition equivalent to unique ergodicity. A measurable function \(V: X\to {\mathbb R}_+\) is Lyapunov for a Markov semigroup \(P_t\) if there exist strictly positive constants \(c_V, \gamma, K_v\) such that \[ P_t V(x) \leq c_V e^{-\gamma t}V(x) + K_V \] for all \(x\in X\) and \(t\geq 0\). A small set for \(P_t\) is a set \(A\) such that there exists \(\delta > 0\) with \(\|P_t(x, \cdot) - P_t(y, \cdot)\|_{TV} \leq 1- \delta\) for any \(x, y\in A\). Harris’ theorem states that, if every level set of a Lyapunov function is small for \(P_{t_*}\) with \(\gamma \cdot t_* > \ln c_V\), then there exists a unique invariant probability measure for the Markov semigroup \(P_t\). The authors prove a Harris-type theorem under slightly generalized conditions. Section 2 is devoted to the proof of Theorem 1.1, the unique ergodicity through asymptotically coupling, and a simple criterion on the convergence of transition probabilities towards an invariant measure in Theorem 2.4.
Section 3 gives applications of the uniqueness and convergence criteria to SDDE, as Theorem 3.1 for at most one invariant measure for SDDE. Furthermore, the authors show in Theorem 3.7 that, for each initial \(\eta \in {\mathcal C}\), the transition probability of the SDDE Markov semigroupconverges to the invariant measure weakly.
Section 4 shows that there is an improved Harris theorem with \(d\)-small set condition and contracting. A set \(A\) is \(d\)-small if there exists \(\varepsilon > 0\) such that \(d(P_t(x, \cdot), P_t(y, \cdot) \leq 1 - \varepsilon\) for all \(x, y\in A\). Having a Lyapunov function \(V\) with \(d\)-small sets is not enough to have the uniqur ergodicity for a Markov semigroup. One needs the distance-like function \(d\) to be contracting. Theorem 4.8 shows that Harris results hold for Markov semigroups with Lyapunov functions such that their level sets are \(d\)-small and \(d\) is contracting.
In the last Section 5, the authors give a distance-like function \(d\) which is contracting and every bounded set is \(d\)-small for the Markov semigroup arising from the SDDE. Hence the results in Section 4 can be applied directly to the SDDE. This gives a positive answer to a long-standing problem of finding natural conditions under which a stochastic delay differential equation admits at most one invariant measure and transition probabilities converge to it.

Reviewer: Weiping Li (Stillwater)

MSC:

60J05	Discrete-time Markov processes on general state spaces
34K50	Stochastic functional-differential equations
37A30	Ergodic theorems, spectral theory, Markov operators

Keywords:

stochastic delay differential equation; invariant measure; Harris’ theorem; weak convergence; spectral gap; asymptotical coupling

Citations:

Zbl 0994.60065

References:

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