Browsing by Author "Hairer, M"

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    Asymptotic coupling and a general form of Harris' theorem with applications to stochastic delay equations
    (Probability Theory and Related Fields, 2011-03-01) Hairer, M; Mattingly, JC; Scheutzow, M
    There are many Markov chains on infinite dimensional spaces whose one-step transition kernels are mutually singular when starting from different initial conditions. We give results which prove unique ergodicity under minimal assumptions on one hand and the existence of a spectral gap under conditions reminiscent of Harris' theorem. The first uses the existence of couplings which draw the solutions together as time goes to infinity. Such "asymptotic couplings" were central to (Mattingly and Sinai in Comm Math Phys 219(3):523-565, 2001; Mattingly in Comm Math Phys 230(3):461-462, 2002; Hairer in Prob Theory Relat Field 124:345-380, 2002; Bakhtin and Mattingly in Commun Contemp Math 7:553-582, 2005) on which this work builds. As in Bakhtin and Mattingly (2005) the emphasis here is on stochastic differential delay equations. Harris' celebrated theorem states that if a Markov chain admits a Lyapunov function whose level sets are "small" (in the sense that transition probabilities are uniformly bounded from below), then it admits a unique invariant measure and transition probabilities converge towards it at exponential speed. This convergence takes place in a total variation norm, weighted by the Lyapunov function. A second aim of this article is to replace the notion of a "small set" by the much weaker notion of a "d-small set," which takes the topology of the underlying space into account via a distance-like function d. With this notion at hand, we prove an analogue to Harris' theorem, where the convergence takes place in a Wasserstein-like distance weighted again by the Lyapunov function. This abstract result is then applied to the framework of stochastic delay equations. In this framework, the usual theory of Harris chains does not apply, since there are natural examples for which there exist no small sets (except for sets consisting of only one point). This gives a solution to the long-standing open problem of finding natural conditions under which a stochastic delay equation admits at most one invariant measure and transition probabilities converge to it. © 2009 Springer-Verlag.
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    Slow energy dissipation in anharmonic oscillator chains
    (Communications on Pure and Applied Mathematics, 2009-08-01) Hairer, M; Mattingly, JC
    We study the dynamic behavior at high energies of a chain of anharmonic oscillators coupled at its ends to heat baths at possibly different temperatures. In our setup, each oscillator is subject to a homogeneous anharmonic pinning potential V 1(qi) = |qi| 2k/2k and harmonic coupling potentials V 2(qi-qi-1) = (qi-q i-1) 2/2 between itself and its nearest neighbors. We consider the case k > 1 when the pinning potential is stronger than the coupling potential. At high energy, when a large fraction of the energy is located in the bulk of the chain, breathers appear and block the transport of energy through the system, thus slowing its convergence to equilibrium. In such a regime, we obtain equations for an effective dynamics by averaging out the fast oscillation of the breather. Using this representation and related ideas, we can prove a number of results. When the chain is of length 3 and k > 3/2, we show that there exists a unique invariant measure. If k > 2 we further show that the system does not relax exponentially fast to this equilibrium by demonstrating that 0 is in the essential spectrum of the generator of the dynamics. When the chain has five or more oscillators and k > 3/2, we show that the generator again has 0 in its essential spectrum. In addition to these rigorous results, a theory is given for the rate of decrease of the energy when it is concentrated in one of the oscillators without dissipation. Numerical simulations are included that confirm the theory. © 2009 Wiley Periodicals, Inc.
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    The strong Feller property for singular stochastic PDEs
    (2016) Hairer, M; Mattingly, J
    We show that the Markov semigroups generated by a large class of singular stochastic PDEs satisfy the strong Feller property. These include for example the KPZ equation and the dynamical $\Phi^4_3$ model. As a corollary, we prove that the Brownian bridge measure is the unique invariant measure for the KPZ equation with periodic boundary conditions.