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 consider a d-dimensional harmonic crystal in contact with a stochastic Langevin type heat bath at each site. The temperatures of the ‘‘exterior’ ’ left and right heat baths are at specified values TL and TR, respectively, while the temperatures of the ‘‘interior’ ’ baths are chosen self-consistently so that there is no average flux of energy between them and the system in the steady state. We prove that this requirement uniquely fixes the temperatures and the self consistent system has a unique steady state. For the infinite system this state is one of local thermal equilibrium. The corresponding heat current satisfies Fourier’s law with a finite positive thermal conductivity which can also be computed using the Green–Kubo formula. For the harmonic chain (d=1) the conductivity agrees with the expression obtained by Bolsterli, Rich, and Visscher in 1970 who first studied this model. In the other limit, d ± 1, the stationary infinite volume heat conductivity behaves as (add) −1 where ad is the coupling to the intermediate reservoirs. We also analyze the effect of having a non-uniform distribution of the heat bath couplings. These results are proven rigorously by controlling the behavior of the correlations in the thermodynamic limit. KEY WORDS: Fourier’s law; harmonic crystal; non-equilibrium systems; thermodynamic limit; Green–Kubo formula.

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}.

Consider the normalized partial sums of a real-valued function F of a Markov chain, φn: = n −1 n−1 F(Φ(k)), n ≥ 1. k=0 The chain {Φ(k) : k ≥ 0} takes values in a general state space X, with transition kernel P, and it is assumed that the Lyapunov drift condition holds: PV ≤ V −W +bIC where V: X → (0, ∞), W: X → [1, ∞), the set C is small, and W dominates F. Under these assumptions, the following conclusions are obtained: (i) It is known that this drift condition is equivalent to the existence of a unique invariant distribution π satisfying π(W) < ∞, and the Law of Large Numbers holds for any function F dominated by W: φn → φ: = π(F), a.s., n → ∞. (ii) The lower error probability defined by P{φn ≤ c}, for c < φ, n ≥ 1, satisfies a large deviation limit theorem when the function F satisfies a monotonicity condition. Under additional minor conditions an exact large deviations expansion is obtained. (iii) If W is near-monotone then control-variates are constructed based on the Lyapunov function V, providing a pair of estimators that together satisfy nontrivial large asymptotics for the lower and upper error probabilities. In an application to simulation of queues it is shown that exact large deviation asymptotics are possible even when the estimator does not satisfy a Central Limit Theorem.

We discuss the Donsker-Varadhan theory of large deviations in the framework of Hamiltonian systems thermostated by a Gaussian stochastic coupling. We derive a general formula for the Donsker-Varadhan large deviation functional for dynamics which satisfy natural properties under time reversal. Next, we discuss the characterization of the stationary state as the solution of a variational principle and its relation to the minimum entropy production principle. Finally, we compute the large deviation functional of the current in the case of a harmonic chain thermostated by a Gaussian stochastic coupling. 1 Introduction. Attempts to study large systems out of equilibrium through fluctuation theory has received a lot of attention in recent years [2, 3, 4, 5, 16, 18, 19, 20]. In a recent series of papers [27, 8, 28, 29, 30], it has been understood that in random systems driven out of equilibrium, the theory of large deviations provides naturally a variational

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.

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

Abstract. In this note we consider a chain of N oscillators, whose ends are in contact with two heat baths at different temperatures. Our main result is the exponential convergence to the unique invariant probability measure (the stationary state). We use the Lyapunov’s function technique of Rey-Bellet and coauthors [11, 8, 13, 12, 4, 5], with different model of heat baths, and adapt these techniques to two new case recently considered in the literature by Bernardin and Olla [2] and Lefevere and

Abstract. The fluctuations in nonequilibrium systems are under intense theoretical and experimental investigation. Topical “fluctuation relations ” describe symmetries of the statistical properties of certain observables, in a variety of models and phenomena. They have been derived in deterministic and, later, in stochastic frameworks. Other results first obtained for stochastic processes, and later considered in deterministic dynamics, describe the temporal evolution of fluctuations. The field has grown beyond expectation: research works and different perspectives are proposed at an ever faster pace. Indeed, understanding fluctuations is important for the emerging theory of nonequilibrium phenomena, as well as for applications, such as those of nanotechnological and biophysical interest. However, the links among the different approaches and the limitations of these approaches are not fully understood. We focus on these issues, providing: a) analysis of the theoretical models; b) discussion of the rigorous mathematical results; c) identification of the physical mechanisms