Brascamp–Lieb-type, weighted Poincaré-type and related analytic inequalities are studied for multidimensional Cauchy distributions and more general κ-concave probability measures (in the hierarchy of convex measures). In analogy with the limiting (infinite-dimensional log-concave) Gaussian model, the weighted inequalities fully describe the measure concentration and large deviation properties of this family of measures. Cheeger-type isoperimetric inequalities are investigated similarly, giving rise to a common weight in the class of concave probability measures under consideration.

This paper addresses the existence and uniqueness of strong solutions to stochastic porous media equations dX−ΔΨ(X) dt=B(X) dW(t) in bounded domains of ℝ^d with Dirichlet boundary conditions. Here Ψ is a maximal monotone graph in ℝ×ℝ (possibly multivalued) with the domain and range all of ℝ. Compared with the existing literature on stochastic porous media equations, no growth condition on Ψ is assumed and the diffusion coefficient Ψ might be multivalued and discontinuous. The latter case is encountered in stochastic models for self-organized criticality or phase transition.

We prove a theorem that reduces bounding the mixing time of a card shuffle to verifying a condition that involves only pairs of cards, then we use it to obtain improved bounds for two previously studied models.

E. Thorp introduced the following card shuffling model in 1973: Suppose the number of cards n is even. Cut the deck into two equal piles. Drop the first card from the left pile or from the right pile according to the outcome of a fair coin flip. Then drop from the other pile. Continue this way until both piles are empty. We obtain a mixing time bound of O(log⁴n). Previously, the best known bound was O(log²⁹n) and previous proofs were only valid for n a power of 2.

We also analyze the following model, called the L-reversal chain, introduced by Durrett: There are n cards arrayed in a circle. Each step, an interval of cards of length at most L is chosen uniformly at random and its order is reversed. Durrett has conjectured that the mixing time is O(max(n, n³/L³)log n). We obtain a bound that is within a factor O(log²n) of this, the first bound within a poly log factor of the conjecture.

We prove an almost sure invariance principle (approximation by d-dimensional Brownian motion) for vector-valued Hölder observables of large classes of nonuniformly hyperbolic dynamical systems. These systems include Axiom A diffeomorphisms and flows as well as systems modeled by Young towers with moderate tail decay rates.

In particular, the position variable of the planar periodic Lorentz gas with finite horizon approximates a two-dimensional Brownian motion.

In this paper we first prove a Clark–Ocone formula for any bounded measurable functional on Poisson space. Then using this formula, under some conditions on the intensity measure of Poisson random measure, we prove a variational representation formula for the Laplace transform of bounded Poisson functionals, which has been conjectured by Dupuis and Ellis [A Weak Convergence Approach to the Theory of Large Deviations (1997) Wiley], p. 122.

We consider a dynamical system described by the differential equation Ẏ_t=−U'(Y_t) with a unique stable point at the origin. We perturb the system by the Lévy noise of intensity ɛ to obtain the stochastic differential equation dX_t^ɛ=−U'(X_t−^ɛ) dt+ɛ dL_t. The process L is a symmetric Lévy process whose jump measure ν has exponentially light tails, ν([u, ∞))∼exp(−u^α), α>0, u→∞. We study the first exit problem for the trajectories of the solutions of the stochastic differential equation from the interval (−1, 1). In the small noise limit ɛ→0, the law of the first exit time σ_x, x∈(−1, 1), has exponential tail and the mean value exhibiting an intriguing phase transition at the critical index α=1, namely, ln Eσ∼ɛ^−α for 0<α<1, whereas ln Eσ∼ɛ⁻¹|ln ɛ|^1−1/α for α>1.

The d-dimensional fractional Brownian motion (FBM for short) B_t=((B_t⁽¹⁾, …, B_t^(d)), t∈ℝ) with Hurst exponent α, α∈(0, 1), is a d-dimensional centered, self-similar Gaussian process with covariance ${\mathbb{E}}[B_{s}^{(i)}B_{t}^{(j)}]=\frac{1}{2}\delta_{i,j}(|s|^{2\alpha}+|t|^{2\alpha}-|t-s|^{2\alpha})$. The long-standing problem of defining a stochastic integration with respect to FBM (and the related problem of solving stochastic differential equations driven by FBM) has been addressed successfully by several different methods, although in each case with a restriction on the range of either d or α. The case α=½ corresponds to the usual stochastic integration with respect to Brownian motion, while most computations become singular when α gets under various threshhold values, due to the growing irregularity of the trajectories as α→0.

We provide here a new method valid for any d and for α>¼ by constructing an approximation Γ(ɛ)_t, ɛ→0, of FBM which allows to define iterated integrals, and then applying the geometric rough path theory. The approximation relies on the definition of an analytic process Γ_z on the cut plane z∈ℂ∖ℝ of which FBM appears to be a boundary value, and allows to understand very precisely the well-known (see [5]) but as yet a little mysterious divergence of Lévy’s area for α→¼.

In this paper we study the tightness of solutions for a family of recursion equations. These equations arise naturally in the study of random walks on tree-like structures. Examples include the maximal displacement of a branching random walk in one dimension and the cover time of a symmetric simple random walk on regular binary trees. Recursion equations associated with the distribution functions of these quantities have been used to establish weak laws of large numbers. Here, we use these recursion equations to establish the tightness of the corresponding sequences of distribution functions after appropriate centering. We phrase our results in a fairly general context, which we hope will facilitate their application in other settings.

We study the sandpile model in infinite volume on ℤ^d. In particular, we are interested in the question whether or not initial configurations, chosen according to a stationary measure μ, are μ-almost surely stabilizable. We prove that stabilizability does not depend on the particular procedure of stabilization we adopt. In d=1 and μ a product measure with density ρ=1 (the known critical value for stabilizability in d=1) with a positive density of empty sites, we prove that μ is not stabilizable.

Furthermore, we study, for values of ρ such that μ is stabilizable, percolation of toppled sites. We find that for ρ>0 small enough, there is a subcritical regime where the distribution of a cluster of toppled sites has an exponential tail, as is the case in the subcritical regime for ordinary percolation.

Let D⊊R^d be an unbounded domain and let B(t) be a Brownian motion in D with normal reflection at the boundary. We study the transcience/recurrence dichotomy, focusing mainly on domains of the form D={(x, z)∈R^l+m) : |z|<H(|x|)}, where d=l+m and H is a sufficiently regular function. This class of domains includes various horn-shaped domains and generalized slab domains.

We consider the real-valued centered Gaussian field on the four-dimensional integer lattice, whose covariance matrix is given by the Green’s function of the discrete Bilaplacian. This is interpreted as a model for a semiflexible membrane. d=4 is the critical dimension for this model. We discuss the effect of a hard wall on the membrane, via a multiscale analysis of the maximum of the field. We use analytic and probabilistic tools to describe the correlation structure of the field.

We consider the asymptotic behavior as n→∞ of the spectra of random matrices of the form $$\frac{1}{\sqrt{n-1}}\sum_{k=1}^{n-1}Z_{nk}\rho_{n}\bigl ((k,k+1)\bigr),$$ where for each n the random variables Z_nk are i.i.d. standard Gaussian and the matrices ρ_n((k, k+1)) are obtained by applying an irreducible unitary representation ρ_n of the symmetric group on {1, 2, …, n} to the transposition (k, k+1) that interchanges k and k+1 [thus, ρ_n((k, k+1)) is both unitary and self-adjoint, with all eigenvalues either +1 or −1]. Irreducible representations of the symmetric group on {1, 2, …, n} are indexed by partitions λ_n of n. A consequence of the results we establish is that if λ_n,1≥λ_n,2≥⋯≥0 is the partition of n corresponding to ρ_n, μ_n,1≥μ_n,2≥⋯≥0 is the corresponding conjugate partition of n (i.e., the Young diagram of μ_n is the transpose of the Young diagram of λ_n), lim_n→∞λ_n,i/n=p_i for each i≥1, and lim_n→∞μ_n,j/n=q_j for each j≥1, then the spectral measure of the resulting random matrix converges in distribution to a random probability measure that is Gaussian with random mean θZ and variance 1−θ², where θ is the constant ∑_ip_i²−∑_jq_j² and Z is a standard Gaussian random variable.

We establish a second-order almost sure limit theorem for the minimal position in a one-dimensional super-critical branching random walk, and also prove a martingale convergence theorem which answers a question of Biggins and Kyprianou [Electron. J. Probab. 10 (2005) 609–631]. Our method applies, furthermore, to the study of directed polymers on a disordered tree. In particular, we give a rigorous proof of a phase transition phenomenon for the partition function (from the point of view of convergence in probability), already described by Derrida and Spohn [J. Statist. Phys. 51 (1988) 817–840]. Surprisingly, this phase transition phenomenon disappears in the sense of upper almost sure limits.

We introduce and study invariant (weighted) transport-kernels balancing stationary random measures on a locally compact Abelian group. The first main result is an associated fundamental invariance property of Palm measures, derived from a generalization of Neveu’s exchange formula. The second main result is a simple sufficient and necessary criterion for the existence of balancing invariant transport-kernels. We then introduce (in a nonstationary setting) the concept of mass-stationarity with respect to a random measure, formalizing the intuitive idea that the origin is a typical location in the mass. The third main result of the paper is that a measure is a Palm measure if and only if it is mass-stationary.