Consider a sequence of i.i.d.\ random variables, where each variable is refreshed (i.e., replaced by an independent variable with the same law) independently, according to a Poisson clock. At any fixed time t, the resulting sequence has the same law as at time 0, but there can be exceptional random times at which certain almost sure properties of the time 0 sequence are violated. We prove that there are no such exceptional times for the law of large numbers and the law of the iterated logarithm, so these laws are dynamically stable. However, there are times at which run lengths are exceptionally long, that is, run tests are dynamically sensitive. We obtain a multifractal analysis of exceptional times for run lengths and for prediction. In particular, starting from an i.i.d. sequence of unbiased random bits, the random set of times t where $\alpha \log_2(n)$ bits among the first n bits can be predicted from their predecessors, has Hausdorff dimension $1-\alpha$ a.s. Finally, we consider simple random walk in the lattice $\Z^d$, and prove that transience is dynamically stable for $d \ge 5$, and dynamically sensitive for $d=3,4$. Moreover, for $d=3,4$, the nonempty random set of exceptional times t where the walk is recurrent has Hausdorff dimension $(4-d)/2$ a.s.

Resolvent $H_{-1}$ norms with respect to simple exclusion processes play an important role in many problems with respect to additive functionals, tagged particles, and hydrodynamics, among other concerns. Here, general translation-invariant finite-range simple exclusion processes with and without a distinguished particle are considered. For the standard system of indistinguishable particles, it is proved that the corresponding $H_{-1}$ norms are equivalent, in a sense, to the $H_{-1}$ norms of a nearest-neighbor system. The same result holds for systems with a distinguished particle in dimensions $d\geq 2$. However, in dimension $d=1$, this equivalence does not hold. An application of the $H_{-1}$ norm equivalence to additive functional variances is also given.

In this paper, we prove a new version of the Birkhoff ergodic theorem (BET) for random variables depending on a parameter (alias integrands). This involves variational convergences, namely epigraphical, hypographical and uniform convergence and requires a suitable definition of the conditional expectation of integrands. We also have to establish the measurability of the epigraphical lower and upper limits with respect to the $\sigma$-field of invariant subsets. From the main result, applications to uniform versions of the BET to sequences of random sets and to the strong consistency of estimators are briefly derived.

We investigate random sequences $(X_n)_{\nin_0}$ with spectral representation based on certain orthogonal polynomials, that is, random sequences that are weakly stationary with respect to polynomial hypergroups. We present various situations where one meets this kind of sequence. The main topic is on the one-step prediction. In particular, it is examined when the mean-squared error tends to zero. For many cases we present a complete solution for the problem of $(X_n)_{\nin_0}$ being asymptotically deterministic.

We consider the Ginzburg--Landau process on the lattice $\mathbb{Z}^d$ whose potential is a bounded perturbation of the Gaussian potential. We prove that the decay rate to equilibrium in the variance sense is $t^{-d/2}$ up to a~logarithmic correction, for any function $u$ with finite triple norm; that is, $|\!|\!| u |\!|\!| \;=\; \sum_{x\in \mathbb{Z}^d} \Vert \partial_{\eta_x} u \Vert_\infty < \infty$.

Consider a one-dimensional exclusion process with finite-range translation-invariant jump rates with nonzero drift. Let the process be stationary with product Bernoulli invariant distribution at density $\rho$. Place a second-class particle initially at the origin. For the case $\rho\neq 1/2$ we show that the time spent by the second-class particle at the origin has finite expectation. This strong transience is then used to prove that variances of additive functionals of local mean-zero functions are diffusive when $\rho\neq 1/2$. As a corollary to previous work, we deduce the invariance principle for these functionals. The main arguments are comparisons of $H_{-1}$ norms, a large deviation estimate for second-class particles and a relation between occupation times of second-class particles, and additive functional variances.

Let $X(t)$ be the symmetric $\alpha$-stable process in $\R$, $\alpha \in (0,2)$, $d \ge 2$. For $f\dvtx (0,1) \to (0,\infty)$ let $D(f)$ be the thorn $\{x \in \R\dvtx x_{1} \in (0,1),\allowbreak |(x_{2},\ldots,x_{d})| < f(x_{1})\}$. We give an integral criterion in terms of $f$ for the existence of a random time $s $ such that $X(t)$ remains in $X(s) + \overline{D}(f)$ for all $t \in [s,s+1)$.

For noncorrelated random variables, we study the rate of approximation of distributions of weighted sums by "typical'' distributions.

In this paper we give explicit representations for Kullback--Leibler information numbers between a priori and a posteriori distributions, when the observations come from a semimartingale. We assume that the distribution of the observed semimartingale is described in terms of the so-called triplet of predictable characteristics. We end by considering the corresponding notions in a model with a fractional noise.

We relate the recurrence and transience of a branching diffusion process on a Riemannian manifold M to some properties of a linear elliptic operator onM (including spectral properties). There is a trade-off between the tendency of the transient Brownian motion to escape and the birth process of the new particles. If the latter has a high enough intensity then it may override the transience of the Brownian motion, leading to the recurrence of the branching process, and vice versa. In the case of a spherically symmetric manifold, the critical intensity of the population growth can be found explicitly.

In this article we show that random walks in random environment on $\mathbb{Z}^d$, $d \ge3$, with transition probabilities which are $\varepsilon$-perturbations of the simple random walk and such that the expectation of the local drift has size bigger than $\varepsilon^\rho $, with $\rho< \frac{5}{2}$, when $d=3$, $\rho< 3$, when $d \ge4$, fulfill the condition (T$^\prime$) introduced by Sznitman [Prob. Theory Related Fields (2002) 122 509-544], when $\varepsilon$ is small. As a result these walks satisfy a law of large numbers with nondegenerate limiting velocity, a central limit theorem and several large deviation controls. In particular, this provides examples of ballistic random walks in random environment which do not satisfy Kalikow's condition in the terminology of Sznitman and Zerner [Ann. Probab. (1999) 27 1851-1869]. An important tool in this work is the effective criterion of Sznitman.

We study a white-noise driven semilinear partial differential equation on the spatial interval $[0,1]$ with Dirichlet boundary condition and with a singular drift of the form $c u^{-3}$, $c>0$. We prove existence and uniqueness of a non-negative continuous adapted solution $u$ on $[0,\infty)\times[0,1]$ for every nonnegative continuous initial datum $x$, satisfying $x(0)=x(1)=0$. We prove that the law $\pi_\delta$ of the Bessel bridge on $[0,1]$ of dimension $\delta>3$ is the unique invariant probability measure of the process $x\mapsto u$, with $c=(\delta-1)(\delta-3)/8$ and, if $\delta\in{\mathbb N}$, that $u$ is the radial part in the sense of Dirichlet forms of the ${\mathbb R}^\delta$-valued solution of a linear stochastic heat equation. An explicit integration by parts formula w.r.t. $\pi_\delta$ is given for all $\delta>3$.

We consider spread-out models of self-avoiding walk, bond percolation, lattice trees and bond lattice animals on ${\mathbb{Z}^d}$, having long finite-range connections, above their upper critical dimensions $d=4$ (self-avoiding walk), $d=6$ (percolation) and $d=8$ (trees and animals). The two-point functions for these models are respectively the generating function for self-avoiding walks from the origin to $x \in {\mathbb{Z}^d}$, the probability of a connection from 0 to x, and the generating function for lattice trees or lattice animals containing 0 and x. We use the lace expansion to prove that for sufficiently spread-out models above the upper critical dimension, the two-point function of each model decays, at the critical point, as a multiple of $|x|^{2-d}$ as $x \to \infty$. We use a new unified method to prove convergence of the lace expansion. The method is based on x-space methods rather than the Fourier transform. Our results also yield unified and simplified proofs of the bubble condition for self-avoiding walk, the triangle condition for percolation, and the square condition for lattice trees and lattice animals, for sufficiently spread-out models above the upper critical dimension.

A version of the SLLN for a large class of means is proved.

Consider the following stochastic differential equation in $\mathbb{R}^d$: \begin{eqnarray*} dX^\varepsilon_t &=& b(X_t^\varepsilon,\xi_{t/\varepsilon})\,dt +\sqrt{\varepsilon} a(X_t^\varepsilon,\xi_{t/\varepsilon})\,dW_t,\\ X^\varepsilon_0 &=& x_0, \end{eqnarray*} where the random environment $(\xi_t)$ is an exponentially ergodic Markov process, independent of the Wiener process $(W_t)$, with invariant probability measure $\pi$, and $\varepsilon$ is some small parameter. In this paper we prove the moderate deviations for the averaging principle of $X^\varepsilon$, that is, deviations of $(X^\varepsilon_t)$ around its limit averaging system $(\bar x_t)$ given by %$\bar{d}$. % $d\bar x_t=\bar b(\bar x_t)\,dt$ and $\bar x_0=x_0$ where $\bar b(x)=\mathbb{E}_\pi(b(x,\cdot))$. More precisely we obtain the large deviation estimation about \[ \Bigl(\eta^\varepsilon_t={X^\varepsilon_t-\bar x_t \over \sqrt{\varepsilon} h(\varepsilon)}\Big)_{t\in[0,1]} \] in the space of continuous trajectories, as $\varepsilon$ decreases to 0, where $h(\varepsilon)$ is some deviation scale satisfying $1\ll h(\varepsilon)\ll\varepsilon^{-1/2}$. Our strategy will be first to show the exponential tightness and then the local moderate deviation principle, which relies on some new method involving a conditional Schilder's theorem and a moderate deviation principle for inhomogeneous integral functionals of Markov processes, previously established by the author.

We study several kinds of large critical percolation clusters in two dimensions. We show that from the microscopic (lattice scale) perspective these clusters can be described by Kesten's incipient infinite cluster (IIC), as was conjectured by Aizenman. More specifically, we establish this for incipient spanning clusters, large clusters in a finite box and the inhomogeneous model of Chayes, Chayes and Durrett. Our results prove the equivalence of several natural definitions of the IIC.

We also show that for any $k \ge 1$ the difference in size between the $k$th and $(k+1)$st largest critical clusters in a finite box goes to infinity in probability as the size of the box goes to infinity. In addition, the distribution of the Chayes--Chayes--Durrett cluster is shown to be singular with respect to the IIC.

We develop a technique for "partially collapsing'' one Markov process to produce another. The state space of the new Markov process is obtained by a pinching operation that identifies points of the original state space via an equivalence relationship. To ensure that the new process is Markovian we need to introduce a randomized twist according to an appropriate probability kernel. Informally, this twist randomizes over the uncollapsed region of the state space when the process leaves the collapsed region. The "Markovianity'' of the new process is ensured by suitable intertwining relationships between the semigroup of the original process and the pinching and twisting operations. We construct the new Markov process, identify its resolvent and transition function and, under some natural assumptions, exhibit a core for its generator. We also investigate its excursion decomposition. We apply our theory to a number of examples, including Walsh's spider and a process similar to one introduced by Sowers in studying stochastic averaging.