We consider a process on $\mathbb{T}^{2}$, which consists of fast motion along the stream lines of an incompressible periodic vector field perturbed by white noise. It gives rise to a process on the graph naturally associated to the structure of the stream lines of the unperturbed flow. It has been shown by Freidlin and Wentzell [Random Perturbations of Dynamical Systems, 2nd ed. Springer, New York (1998)] and [Mem. Amer. Math. Soc. 109 (1994)] that if the stream function of the flow is periodic, then the corresponding process on the graph weakly converges to a Markov process. We consider the situation where the stream function is not periodic, and the flow (when considered on the torus) has an ergodic component of positive measure. We show that if the rotation number is Diophantine, then the process on the graph still converges to a Markov process, which spends a positive proportion of time in the vertex corresponding to the ergodic component of the flow.

We develop a general method to prove the existence of spectral gaps for Markov semigroups on Banach spaces. Unlike most previous work, the type of norm we consider for this analysis is neither a weighted supremum norm nor an Ł^p-type norm, but involves the derivative of the observable as well and hence can be seen as a type of 1-Wasserstein distance. This turns out to be a suitable approach for infinite-dimensional spaces where the usual Harris or Doeblin conditions, which are geared toward total variation convergence, often fail to hold. In the first part of this paper, we consider semigroups that have uniform behavior which one can view as the analog of Doeblin’s condition. We then proceed to study situations where the behavior is not so uniform, but the system has a suitable Lyapunov structure, leading to a type of Harris condition. We finally show that the latter condition is satisfied by the two-dimensional stochastic Navier–Stokes equations, even in situations where the forcing is extremely degenerate. Using the convergence result, we show that the stochastic Navier–Stokes equations’ invariant measures depend continuously on the viscosity and the structure of the forcing.

In this paper, we propose a new notion of Forward–Backward Martingale Problem (FBMP), and study its relationship with the weak solution to the forward–backward stochastic differential equations (FBSDEs). The FBMP extends the idea of the well-known (forward) martingale problem of Stroock and Varadhan, but it is structured specifically to fit the nature of an FBSDE. We first prove a general sufficient condition for the existence of the solution to the FBMP. In the Markovian case with uniformly continuous coefficients, we show that the weak solution to the FBSDE (or equivalently, the solution to the FBMP) does exist. Moreover, we prove that the uniqueness of the FBMP (whence the uniqueness of the weak solution) is determined by the uniqueness of the viscosity solution of the corresponding quasilinear PDE.

The martingale method is used to establish concentration inequalities for a class of dependent random sequences on a countable state space, with the constants in the inequalities expressed in terms of certain mixing coefficients. Along the way, bounds are obtained on martingale differences associated with the random sequences, which may be of independent interest. As applications of the main result, concentration inequalities are also derived for inhomogeneous Markov chains and hidden Markov chains, and an extremal property associated with their martingale difference bounds is established. This work complements and generalizes certain concentration inequalities obtained by Marton and Samson, while also providing different proofs of some known results.

The present article is devoted to a fine study of the convergence of renormalized weighted quadratic and cubic variations of a fractional Brownian motion B with Hurst index H. In the quadratic (resp. cubic) case, when H<1/4 (resp. H<1/6), we show by means of Malliavin calculus that the convergence holds in L² toward an explicit limit which only depends on B. This result is somewhat surprising when compared with the celebrated Breuer and Major theorem.

Let ξ be a Dawson–Watanabe superprocess in ℝ^d such that ξ_t is a.s. locally finite for every t≥0. Then for d≥2 and fixed t>0, the singular random measure ξ_t can be a.s. approximated by suitably normalized restrictions of Lebesgue measure to the ɛ-neighborhoods of supp ξ_t. When d≥3, the local distributions of ξ_t near a hitting point can be approximated in total variation by those of a stationary and self-similar pseudo-random measure ξ̃. By contrast, the corresponding distributions for d=2 are locally invariant. Further results include improvements of some classical extinction criteria and some limiting properties of hitting probabilities. Our main proofs are based on a detailed analysis of the historical structure of ξ.

Let $(\Omega,\mathcal{F},P)$ be a probability space and $\mathcal{N}$ the class of those $F\in\mathcal{F}$ satisfying P(F)∈{0, 1}. For each $\mathcal{G}\subset\mathcal{F}$, define $\overline{\mathcal{G}}=\sigma(\mathcal{G}\cup\mathcal {N})$. Necessary and sufficient conditions for $\overline{\mathcal{A}}\cap\overline{\mathcal{B}}=\overline {\mathcal {A}\cap\mathcal{B}}$, where $\mathcal{A},\mathcal{B}\subset\mathcal{F}$ are sub-σ-fields, are given. These conditions are then applied to the (two-component) Gibbs sampler. Suppose X and Y are the coordinate projections on $(\Omega,\mathcal{F})=(\mathcal{X}\times\mathcal{Y},\mathcal {U}\otimes \mathcal{V})$ where $(\mathcal{X},\mathcal{U})$ and $(\mathcal{Y},\mathcal{V})$ are measurable spaces. Let (X_n, Y_n)_n≥0 be the Gibbs chain for P. Then, the SLLN holds for (X_n, Y_n) if and only if $\overline{\sigma(X)}\cap\overline{\sigma(Y)}=\mathcal{N}$, or equivalently if and only if P(X∈U)P(Y∈V)=0 whenever $U\in\mathcal{U}$, $V\in\mathcal{V}$ and P(U×V)=P(U^c×V^c)=0. The latter condition is also equivalent to ergodicity of (X_n, Y_n), on a certain subset S₀⊂Ω, in case $\mathcal{F}=\mathcal{U}\otimes\mathcal{V}$ is countably generated and P absolutely continuous with respect to a product measure.

We consider a real-valued path; it is possible to associate a tree to this path, and we explore the relations between the tree, the properties of p-variation of the path, and integration with respect to the path. In particular, the fractal dimension of the tree is estimated from the variations of the path, and Young integrals with respect to the path, as well as integrals from the rough paths theory, are written as integrals on the tree. Examples include some stochastic paths such as martingales, Lévy processes and fractional Brownian motions (for which an estimator of the Hurst parameter is given).

Let X=(X₁, X₂, …) be a nondeterministic infinite exchangeable sequence with values in {0, 1}. We show that X is Hoeffding decomposable if, and only if, X is either an i.i.d. sequence or a Pólya sequence. This completes the results established in Peccati [Ann. Probab. 32 (2004) 1796–1829]. The proof uses several combinatorial implications of the correspondence between Hoeffding decomposability and weak independence. Our results must be compared with previous characterizations of i.i.d. and Pólya sequences given by Hill, Lane and Sudderth [Ann. Probab. 15 (1987) 1586–1592] and Diaconis and Ylvisaker [Ann. Statist. 7 (1979) 269–281]. The final section contains a partial characterization of Hoeffding decomposable sequences with values in a set with more than two elements.

Pathwise uniqueness holds for the Skorokhod stochastic differential equation in C^1+γ domains in ℝ^d for γ>1/2 and d≥3.

Recent developments in quantum physics make heavy use of so-called “quantum trajectories.” Mathematically, this theory gives rise to “stochastic Schrödinger equations,” that is, perturbation of Schrödinger-type equations under the form of stochastic differential equations. But such equations are in general not of the usual type as considered in the literature. They pose a serious problem in terms of justifying the existence and uniqueness of a solution, justifying the physical pertinence of the equations. In this article we concentrate on a particular case: the diffusive case, for a two-level system. We prove existence and uniqueness of the associated stochastic Schrödinger equation. We physically justify the equations by proving that they are a continuous-time limit of a concrete physical procedure for obtaining a quantum trajectory.

We consider transient random walks on a strip in a random environment. The model was introduced by Bolthausen and Goldsheid [Comm. Math. Phys. 214 (2000) 429–447]. We derive a strong law of large numbers for the random walks in a general ergodic setup and obtain an annealed central limit theorem in the case of uniformly mixing environments. In addition, we prove that the law of the “environment viewed from the position of the walker” converges to a limiting distribution if the environment is an i.i.d. sequence.

We consider a random field ϕ:{1, …, N}→ℝ as a model for a linear chain attracted to the defect line ϕ=0, that is, the x-axis. The free law of the field is specified by the density exp(−∑_iV(Δϕ_i)) with respect to the Lebesgue measure on ℝ^N, where Δ is the discrete Laplacian and we allow for a very large class of potentials V(⋅). The interaction with the defect line is introduced by giving the field a reward ɛ≥0 each time it touches the x-axis. We call this model the pinning model. We consider a second model, the wetting model, in which, in addition to the pinning reward, the field is also constrained to stay nonnegative.

We show that both models undergo a phase transition as the intensity ɛ of the pinning reward varies: both in the pinning (a=p) and in the wetting (a=w) case, there exists a critical value ɛ_c^a such that when ɛ>ɛ_c^a the field touches the defect line a positive fraction of times (localization), while this does not happen for ɛ<ɛ_c^a (delocalization). The two critical values are nontrivial and distinct: 0<ɛ_c^p<ɛ_c^w<∞, and they are the only nonanalyticity points of the respective free energies. For the pinning model the transition is of second order, hence the field at ɛ=ɛ_c^p is delocalized. On the other hand, the transition in the wetting model is of first order and for ɛ=ɛ_c^w the field is localized. The core of our approach is a Markov renewal theory description of the field.