Abstract. We prove the Gallavotti-Cohen fluctuation theorem for a model of heat conduction through a chain of anharmonic oscillators coupled to two Hamiltonian reservoirs at different temperatures. 1

Diffusion models arising in analysis of large biochemical models and other complex systems are typically far too complex for exact solution, or even meaningful simulation. The purpose of this paper is to develop foundations for model reduction, and new modeling techniques for diffusion models. These foundations are all based upon recent spectral theory of Markov processes. The main assumption imposed is V-uniform ergodicity of the process. This is equivalent to any common formulation of exponential ergodicity, and is known to be far weaker than the Donsker-Varadahn conditions in large deviations theory. Under this assumption it is shown that the associated semigroup admits a spectral gap in a weighted L∞-norm, and real eigenfunctions provide a decomposition of the state space into ‘almost’-absorbing subsets. It is shown that the process mixes rapidly in each of these subsets prior to exiting, and that the conditional distributions of exit times are approximately exponential. These results represent a significant expansion of the classical Wentzell–Freidlin theory. In particular, the results require no special structure beyond geometric ergodicity; reversibility is not assumed; and meaningful conclusions can be drawn even for models with significant variability.

In this paper we continue the investigation of the spectral theory and exponential asymptotics of primarily discrete-time Markov processes, following Kontoyiannis and Meyn [32]. We introduce a new family of nonlinear Lyapunov drift criteria, which characterize distinct subclasses of geometrically ergodic Markov processes in terms of simple inequalities for the nonlinear generator. We concentrate primarily on the class of multiplicatively regular Markov processes, which are characterized via simple conditions similar to (but weaker than) those of Donsker-Varadhan. For any such process Φ = {Φ(t)} with transition kernel P on a general state space X, the following are obtained. Spectral Theory: For a large class of (possibly unbounded) functionals F: X → C, the kernel ̂ P (x, dy) = e F (x) P (x, dy) has a discrete spectrum in an appropriately defined Banach space. It follows that there exists a “maximal ” solution (λ, ˇ f) to the multiplicative Poisson equation, defined as the eigenvalue problem ̂ P ˇ f = λ ˇ f. The functional Λ(F) = log(λ) is convex, smooth, and its convex dual Λ ∗ is convex, with compact sublevel sets.

We study hypoelliptic operators with polynomially bounded coefficients that are of the form K = i X i + X0 + f , where the X j denote first order differential operators, f is a function with at most polynomial growth, and X i denotes the formal adjoint of X i in L . For any # > 0 we show that an inequality of the form C(#u#0,# + + iy)u#0,0 ) holds for suitable # and C which are independent of R, in weighted Sobolev spaces (the first index is the derivative, and the second the growth). We apply this result to the Fokker-Planck operator for an anharmonic chain of oscillators coupled to two heat baths. Using a method of H erau and Nier [HN02], we conclude that its spectrum lies in a cusp {x+ iy # |y| c, # (0, 1], c R}.

We consider the dynamics of a periodic chain of N coupled overdamped particles under the influence of noise, in the limit of large N. Each particle is subjected to a bistable local potential, to a linear coupling with its nearest neighbours, and to an independent source of white noise. For strong coupling (of the order N 2), the system synchronises, in the sense that all oscillators assume almost the same position in their respective local potential most of the time. In a previous paper, we showed that the transition from strong to weak coupling involves a sequence of symmetry-breaking bifurcations of the system’s stationary configurations, and analysed in particular the behaviour for coupling intensities slightly below the synchronisation threshold, for arbitrary N. Here we describe the behaviour for any positive coupling intensity γ of order N 2, provided the particle number N is sufficiently large (as a function of γ/N 2). In particular, we determine the transition time between synchronised states, as well as the shape of the “critical droplet ” to leading order in 1/N. Our techniques involve the control of the exact number of periodic orbits of a near-integrable twist map, allowing us to give a detailed description of the system’s potential landscape, in which the metastable behaviour is encoded.

Abstract. We consider a stochastic Klein-Gordon wave equation modeling heat flow in a linear field that is coupled to thermal reservoirs at different temperatures. We discuss, in a perturbative context, the approach to a stationary, non-equilibrium state, and show that the state is supported on field configurations which are Hölder continuous, with any exponent less than 1/2. We determine the heat flux to lowest order in perturbation theory. 1.

Diffusion models arising in analysis of real world systems are typically far too complex for exact solution, or even meaningful simulation. The purpose of this paper is to develop foundations for model reduction, and new modeling techniques for diffusion models. Based on the main assumption of V -uniform ergodicity of the diffusion process it is shown that real eigenfunctions provide a decomposition of the state space into so-called metastable sets. We give a novel definition of metastability via exit rates which seems to be promising for a algorithmic identification of metastable sets even for large scale systems.

Abstract not found

Isotropic hypoellipticity and trend to the equilibrium for the Fokker-Planck equation with high degree potential

Abstract — The purpose of this paper is to develop methods for model reduction for diffusion processes that exhibit cyclic behavior. For this purpose we extend techniques based on the spectral theory of Markov processes to the case of complex spectra. The main idea is to augment the state process for the diffusion with a clock process. For each complex eigenvalue for the original diffusion there exists a real eigenvalue for the augmented process. Results concerning metastability (or quasi-stationarity) are then applied to the augmented process. For the special case of a linear diffusion in two dimensions, this is analogous to analyzing the process in a rotating coordinate frame. The results are illustrated through a linear diffusion, and an empirical model of combustion dynamics. I.