We consider the random walk on supercritical percolation clusters in $\mathbb{Z}^d$. Previous papers have obtained Gaussian heat kernel bounds, and a.s. invariance principles for this process. We show how this information leads to a parabolic Harnack inequality, a local limit theorem and estimates on the Green's function.

We give simple conditions that ensure exponential forgetting of the initial conditions of the filter for general state-space hidden Markov chain. The proofs are based on the coupling argument applied to the posterior Markov kernels. These results are useful both for filtering hidden Markov models using approximation methods (e.g., particle filters) and for proving asymptotic properties of estimators. The results are general enough to cover models like the Gaussian state space model, without using the special structure that permits the application of the Kalman filter.

Let $D$ be a bounded Lipschitz domain in $R^n$ with $n\geq 2$ and $\tau_D$ be the first exit time from $D$ by Brownian motion on $R^n$. In the first part of this paper, we are concerned with sharp estimates on the expected exit time $E_x [ \tau_D]$. We show that if $D$ satisfies a uniform interior cone condition with angle $\theta \in ( \cos^{-1}(1/\sqrt{n}), \pi )$, then $c_1 \varphi_1(x) \leq E_x [\tau_D] \leq c_2 \varphi_1 (x)$ on $D$. Here $\varphi_1$ is the first positive eigenfunction for the Dirichlet Laplacian on $D$. The above result is sharp as we show that if $D$ is a truncated circular cone with angle $\theta < \cos^{-1}(1/\sqrt{n})$, then the upper bound for $E_x [\tau_D]$ fails. These results are then used in the second part of this paper to investigate whether positive solutions of the semilinear equation $\Delta u = u^{p}$ in $ D,$ $p\in R$, that vanish on an open subset $\Gamma \subset \partial D$ decay at the same rate as $\varphi_1$ on $\Gamma$.

We consider random selection processes of weighted elements in an arbitrary set. Their conditional distributions are shown to be a generalization of the hypergeometric distribution, while the marginal distributions can always be chosen as generalized binomial distributions. Then we propose sufficient conditions on the weight function ensuring that the marginal distributions are necessarily of the generalized binomial form. In these cases, the corresponding indicator random variables are conditionally independent (as in the classical De Finetti theorem) though they are neither exchangeable nor identically distributed.

We study the random graph obtained by random deletion of vertices or edges from a random graph with given vertex degrees. A simple trick of exploding vertices instead of deleting them, enables us to derive results from known results for random graphs with given vertex degrees. This is used to study existence of giant component and existence of k-core. As a variation of the latter, we study also bootstrap percolation in random regular graphs. We obtain both simple new proofs of known results and new results. An interesting feature is that for some degree sequences, there are several or even infinitely many phase transitions for the k-core.

We study a one dimensional constant threshold model in continuous time. Its dynamics have two parameters, the range $n$ and the threshold $v$. An unoccupied site $x$ becomes occupied at rate 1 as soon as there are at least $v$ occupied sites in $[x-n, x+n]$. As n goes to infinity and $v$ is kept fixed, the dynamics can be approximated by a continuous space version, which has an explicit invariant measure at the front. This allows us to prove that the speed of propagation is asymptoticaly $n^2/2v$.

We propose a simple model for the behaviour of longterm investors on a stock market. It consists of three particles that represent the stock's current price and the buyers', respectively sellers', opinion about the right trading price. As time evolves, both groups of traders update their opinions with respect to the current price. The speed of updating is controled by a parameter; the price process is described by a geometric Brownian motion. We consider the market's stability in terms of the distance between the buyers' and sellers' opinion, and prove that the distance process is recurrent/transient in dependence on the parameter.

We prove upper and lower bounds on the free energy of the Sherrington-Kirkpatrick model with multidimensional spins in terms of variational inequalities. The bounds are based on a multidimensional extension of the Parisi functional. We generalise and unify the comparison scheme of Aizenman, Sims and Starr and the one of Guerra involving the GREM-inspired processes and Ruelle's probability cascades. For this purpose, an abstract quenched large deviations principle of the Gärtner-Ellis type is obtained. We derive Talagrand's representation of Guerra's remainder term for the Sherrington-Kirkpatrick model with multidimensional spins. The derivation is based on well-known properties of Ruelle's probability cascades and the Bolthausen-Sznitman coalescent. We study the properties of the multidimensional Parisi functional by establishing a link with a certain class of semi-linear partial differential equations. We embed the problem of strict convexity of the Parisi functional in a more general setting and prove the convexity in some particular cases which shed some light on the original convexity problem of Talagrand. Finally, we prove the Parisi formula for the local free energy in the case of multidimensional Gaussian a priori distribution of spins using Talagrand's methodology of a priori estimates.

We investigate the infinitely many demes limit of the genealogy of a sample of individuals from a subdivided population that experiences sporadic mass extinction events. By exploiting a separation of time scales that occurs within a class of structured population models generalizing Wright's island model, we show that as the number of demes tends to infinity, the limiting form of the genealogy can be described in terms of the alternation of instantaneous scattering phases that depend mainly on local demographic processes, and extended collecting phases that are dominated by global processes. When extinction and recolonization events are local, the genealogy is described by Kingman's coalescent, and the scattering phase influences only the overall rate of the process. In contrast, if the demes left vacant by a mass extinction event are recolonized by individuals emerging from a small number of demes, then the limiting genealogy is a coalescent process with simultaneous multiple mergers (a $\Xi$-coalescent). In this case, the details of the within-deme population dynamics influence not only the overall rate of the coalescent process, but also the statistics of the complex mergers that can occur within sample genealogies. These results suggest that the combined effects of geography and disturbance could play an important role in producing the unusual patterns of genetic variation documented in some marine organisms with high fecundity.

Infinite sums of i.i.d. random variables discounted by a multiplicative random walk are called perpetuities and have been studied by many authors. The present paper provides a log-type moment result for such random variables under minimal conditions which is then utilized for the study of related moments of a.s. limits of certain martingales associated with the supercritical branching random walk. The connection arises upon consideration of a size-biased version of the branching random walk originally introduced by Lyons. As a by-product, necessary and sufficient conditions for uniform integrability of these martingales are provided in the most general situation which particularly means that the classical (LlogL)-condition is not always needed.

We consider the Dirichlet form given by $$ {\cal E}(f,f) = \frac{1}{2}\int_{R^d}\sum_{i,j=1}^d a_{ij}(x)\frac{\partial f(x)}{\partial x_i} \frac{\partial f(x)}{\partial x_j} dx$$ $$ + \int_{R^d \times R^d} (f(y)-f(x))^2J(x,y)dxdy.$$ Under the assumption that the ${a_{ij}}$ are symmetric and uniformly elliptic and with suitable conditions on $J$, the nonlocal part, we obtain upper and lower bounds on the heat kernel of the Dirichlet form. We also prove a Harnack inequality and a regularity theorem for functions that are harmonic with respect to $\cal E$.

We prove that the Itô map, that is the map that gives the solution of a differential equation controlled by a rough path of finite $p$-variation with $p\in [2,3)$ is locally Lipschitz continuous in all its arguments and we give some sufficient conditions for global existence for non-bounded vector fields.

The paper is concerned with the classical occupancy scheme in which balls are thrown independently into infinitely many boxes, with given probability of hitting each of the boxes. We establish joint normal approximation, as the number of balls goes to infinity, for the numbers of boxes containing any fixed number of balls, standardized in the natural way, assuming only that the variances of these counts all tend to infinity. The proof of this approximation is based on a de-Poissonization lemma. We then review sufficient conditions for the variances to tend to infinity. Typically, the normal approximation does not mean convergence. We show that the convergence of the full vector of counts only holds under a condition of regular variation, thus giving a complete characterization of possible limit correlation structures.

We study a variant of bootstrap percolation in which growth is restricted to a single active cluster. Initially there is a single active site at the origin, while other sites of $\mathbb{Z}^2$ are independently occupied with small probability $p$, otherwise empty. Subsequently, an empty site becomes active by contact with two or more active neighbors, and an occupied site becomes active if it has an active site within distance 2. We prove that the entire lattice becomes active with probability $\exp [\alpha(p)/p]$, where $\alpha(p)$ is between $-\pi^2/9+c\sqrt p$ and $-\pi^2/9+C\sqrt p(\log p^{-1})^3$. This corrects previous numerical predictions for the scaling of the correction term.

We introduce a simple tree growth process that gives rise to a new two-parameter family of discrete fragmentation trees that extends Ford's alpha model to multifurcating trees and includes the trees obtained by uniform sampling from Duquesne and Le Gall's stable continuum random tree. We call these new trees the alpha-gamma trees. In this paper, we obtain their splitting rules, dislocation measures both in ranked order and in size-biased order, and we study their limiting behaviour.

We consider random walks in Dirichlet random environment. Since the Dirichlet distribution is not uniformly elliptic, the annealed integrability of the exit time out of a given finite subset is a non-trivial question. In this paper we provide a simple and explicit equivalent condition for the integrability of Green functions and exit times on any finite directed graph. The proof relies on a quotienting procedure allowing for an induction argument on the cardinality of the graph. This integrability problem arises in the definition of Kalikow auxiliary random walk. Using a particular case of our condition, we prove a refined version of the ballisticity criterion given by Enriquez and Sabot.

We show that quaternionic Gaussian random variables satisfy a generalization of the Wick formula for computing the expected value of products in terms of a family of graphical enumeration problems. When applied to the quaternionic Wigner and Wishart families of random matrices the result gives the duality between moments of these families and the corresponding real Wigner and Wishart families.

We study the asymptotic behavior of solutions of semilinear PDEs. Neither periodicity nor ergodicity will be assumed. On the other hand, we assume that the coecients have averages in the Cesaro sense. In such a case, the averaged coecients could be discontinuous. We use a probabilistic approach based on weak convergence of the associated backward stochastic dierential equation (BSDE) in the Jakubowski $S$-topology to derive the averaged PDE. However, since the averaged coecients are discontinuous, the classical viscosity solution is not dened for the averaged PDE. We then use the notion of "$L_p$-viscosity solution" introduced in [7]. The existence of $L_p$-viscosity solution to the averaged PDE is proved here by using BSDEs techniques.

We prove a comparison theorem and maximum principle for a local solution of quasi-linear parabolic stochastic PDEs, similar to the well known results in the deterministic case. The proofs are based on a version of Ito's formula and estimates for the positive part of a local solution which is non-positive on the lateral boundary. Moreover we shortly indicate how these results generalize for Burgers type SPDEs

For a much-studied model of random copolymer at a selective interface we prove that the slope of the critical curve in the weak-disorder limit is strictly smaller than 1, which is the value given by the annealed inequality. The proof is based on a coarse-graining procedure, combined with upper bounds on the fractional moments of the partition function.

We consider nonlinear parabolic SPDEs of the form $\partial_t u={\cal L} u + \sigma(u)\dot w$, where $\dot w$ denotes space-time white noise, $\sigma:R\to R$ is [globally] Lipschitz continuous, and $\cal L$ is the $L^2$-generator of a L'evy process. We present precise criteria for existence as well as uniqueness of solutions. More significantly, we prove that these solutions grow in time with at most a precise exponential rate. We establish also that when $\sigma$ is globally Lipschitz and asymptotically sublinear, the solution to the nonlinear heat equation is ``weakly intermittent,'' provided that the symmetrization of $\cal L$ is recurrent and the initial data is sufficiently large. Among other things, our results lead to general formulas for the upper second-moment Liapounov exponent of the parabolic Anderson model for $\cal L$ in dimension $(1+1)$. When ${\cal L}=\kappa\partial_{xx}$ for $\kappa>0$, these formulas agree with the earlier results of statistical physics (Kardar (1987), Krug and Spohn (1991), Lieb and Liniger (1963)), and also probability theory (Bertini and Cancrini (1995), Carmona and Molchanov (1994)) in the two exactly-solvable cases. That is when $u_0=\delta_0$ or $u_0\equiv 1$; in those cases the moments of the solution to the SPDE can be computed (Bertini and Cancrini (1995)).

We consider a Random Walk in Random Environment (RWRE) moving in an i.i.d. random field of obstacles. When the particle hits an obstacle, it disappears with a positive probability. We obtain quenched and annealed bounds on the tails of the survival time in the general <i>d</i>-dimensional case. We then consider a simplified one-dimensional model (where transition probabilities and obstacles are independent and the RWRE only moves to neighbour sites), and obtain finer results for the tail of the survival time. In addition, we study also the "mixed" probability measures (quenched with respect to the obstacles and annealed with respect to the transition probabilities and vice-versa) and give results for tails of the survival time with respect to these probability measures. Further, we apply the same methods to obtain bounds for the tails of hitting times of Branching Random Walks in Random Environment (BRWRE).

We study the correlation decay and the expected maximal increment (Burkholder-Davis-Gundy type inequalities) of the exponential process determined by a fractional Ornstein-Uhlenbeck process. The method is to apply integration by parts formula on integral representations of fractional Ornstein-Uhlenbeck processes, and also to use Slepian's inequality. As an application, we attempt Kahane's T-martingale theory based on our exponential process which is shown to be of long memory.

We discuss a d-dimensional version (for làdlàg optional processes) of a duality result by Meyer (1976) between {bounded} càdlàg adapted processes and random measures. We show that it allows to establish, in a very natural way, a dual representation for the set of initial endowments which allow to super-hedge a given American claim in a continuous time model with proportional transaction costs. It generalizes a previous result of Bouchard and Temam (2005) who considered a discrete time setting. It also completes the very recent work of Denis, De Vallière and Kabanov (2008) who studied càdlàg American claims and used a completely different approach.

Necessary conditions for the existence of a maximal Markovian coupling of diffusion processes are studied. A sufficient condition described as a global symmetry of the processes is revealed to be necessary for the Brownian motion on a Riemannian homogeneous space. As a result, we find many examples of a diffusion process which admits no maximal Markovian coupling. As an application, we find a Markov chain which admits no maximal Markovian coupling for specified starting points.

For any continuous zero-mean random variable $X$, a reciprocating function $r$ is constructed, based only on the distribution of $X$, such that the conditional distribution of $X$ given the (at-most-)two-point set $\{X,r(X)\}$ is the zero-mean distribution on this set; in fact, a more general construction without the continuity assumption is given in this paper, as well as a large variety of other related results, including characterizations of the reciprocating function and modeling distribution asymmetry patterns. The mentioned disintegration of zero-mean r.v.'s implies, in particular, that an arbitrary zero-mean distribution is represented as the mixture of two-point zero-mean distributions; moreover, this mixture representation is most symmetric in a variety of senses. Somewhat similar representations - of any probability distribution as the mixture of two-point distributions with the same skewness coefficient (but possibly with different means) - go back to Kolmogorov; very recently, Aizenman et al. further developed such representations and applied them to (anti-)concentration inequalities for functions of independent random variables and to spectral localization for random Schroedinger operators. One kind of application given in the present paper is to construct certain statistical tests for asymmetry patterns and for location without symmetry conditions. Exact inequalities implying conservative properties of such tests are presented. These developments extend results established earlier by Efron, Eaton, and Pinelis under a symmetry condition.

We consider competing particle systems in $\mathbb{R}^d$, i.e. random locally finite upper bounded configurations of points in $\mathbb{R}^d$ evolving in discrete time steps. In each step i.i.d. increments are added to the particles independently of the initial configuration and the previous steps. Ruzmaikina and Aizenman characterized quasi-stationary measures of such an evolution, i.e. point processes for which the joint distribution of the gaps between the particles is invariant under the evolution, in case $d=1$ and restricting to increments having a density and an everywhere finite moment generating function. We prove corresponding versions of their theorem in dimension $d=1$ for heavy-tailed increments in the domain of attraction of a stable law and in dimension $d\geq 1$ for lattice type increments with an everywhere finite moment generating function. In all cases we only assume that under the initial configuration no two particles are located at the same point. In addition, we analyze the attractivity of quasi-stationary Poisson point processes in the space of all Poisson point processes with almost surely infinite, locally finite and upper bounded configurations.

We give new exponential inequalities and Gaussian approximation results for sums of weakly dependent variables. These results lead to generalizations of Bernstein and Hoeffding inequalities, where an extra control term is added; this term contains conditional moments of the variables.

We obtain upper bounds on the spectral gap of Markov chains constructed by parallel and simulated tempering, and provide a set of sufficient conditions for torpid mixing of both techniques. Combined with the results of Woodard, Schmidler and Huber (2009), these results yield a two-sided bound on the spectral gap of these algorithms. We identify a persistence property of the target distribution, and show that it can lead unexpectedly to slow mixing that commonly used convergence diagnostics will fail to detect. For a multimodal distribution, the persistence is a measure of how ``spiky'', or tall and narrow, one peak is relative to the other peaks of the distribution. We show that this persistence phenomenon can be used to explain the torpid mixing of parallel and simulated tempering on the ferromagnetic mean-field Potts model shown previously. We also illustrate how it causes torpid mixing of tempering on a mixture of normal distributions with unequal covariances in R^M, a previously unknown result with relevance to statistical inference problems. More generally, anytime a multimodal distribution includes both very narrow and very wide peaks of comparable probability mass, parallel and simulated tempering are shown to mix slowly.

The Brownian net, which has recently been introduced by Sun and Swart [16], and independently by Newman, Ravishankar and Schertzer [13], generalizes the Brownian web by allowing branching. In this paper, we study the structure of the Brownian net in more detail. In particular, we give an almost sure classification of each point in $\mathbb{R}^2$ according to the configuration of the Brownian net paths entering and leaving the point. Along the way, we establish various other structural properties of the Brownian net.

It has been proved by Bertoin and Caballero cite{BC2002} that a $1/\alpha$-increasing self-similar Markov process $X$ is such that $t^{-1/\alpha}X(t)$ converges weakly, as $t\to\infty,$ to a degenerate random variable whenever the subordinator associated to it via Lamperti's transformation has infinite mean. Here we prove that $\log(X(t)/t^{1/\alpha})/\log(t)$ converges in law to a non-degenerate random variable if and only if the associated subordinator has Laplace exponent that varies regularly at $0.$ Moreover, we show that $\liminf_{t\to\infty}\log(X(t))/\log(t)=1/\alpha,$ a.s. and provide an integral test for the upper functions of $\{\log(X(t)), t\geq 0\}.$ Furthermore, results concerning the rate of growth of the random clock appearing in Lamperti's transformation are obtained. In particular, these allow us to establish estimates for the left tail of some exponential functionals of subordinators. Finally, some of the implications of these results in the theory of self-similar fragmentations are discussed.

We show that Zhang's sandpile model $(N, [a, b])$ on $N$ sites and with uniform additions on $[a,b]$ has a unique stationary measure for all $0\leq a < b\leq 1$. This generalizes earlier results of cite{anne} where this was shown in some special cases. We define the infinite volume Zhang's sandpile model in dimension $d\geq1$, in which topplings occur according to a Markov toppling process, and we study the stabilizability of initial configurations chosen according to some measure $mu$. We show that for a stationary ergodic measure $\mu$ with density $\rho$, for all $\rho < \frac{1}{2}$, $\mu$ is stabilizable; for all $\rho\geq 1$, $\mu$ is not stabilizable; for $\frac{1}{2}\leq \rho<1$, when $\rho$ is near to $\frac{1}{2}$ or $1$, both possibilities can occur.

By using a change of scale and space, we study a class of stochastic differential equations (SDEs) whose solutions are drift--perturbed and exhibit asymptotic behaviour similar to standard Brownian motion. In particular sufficient conditions ensuring that these processes obey the Law of the Iterated Logarithm (LIL) are given. Ergodic--type theorems on the average growth of these non-stationary processes, which also depend on the asymptotic behaviour of the drift coefficient, are investigated. We apply these results to inefficient financial market models. The techniques extend to certain classes of finite--dimensional equation.

We consider a class of interacting particle systems with values in $[0,∞)^{\mathbb{Z}^d}$, of which the binary contact path process is an example. For $d \geq 3$ and under a certain square integrability condition on the total number of the particles, we prove a central limit theorem for the density of the particles, together with upper bounds for the density of the most populated site and the replica overlap.

In this paper, we give estimates of ideal or minimal distances between the distribution of the normalized partial sum and the limiting Gaussian distribution for stationary martingale difference sequences or stationary sequences satisfying projective criteria. Applications to functions of linear processes and to functions of expanding maps of the interval are given.

We give a new proof of a result of Kenyon that the growth exponent for loop-erased random walks in two dimensions is 5/4. The proof uses the convergence of LERW to Schramm-Loewner evolution with parameter 2, and is valid for irreducible bounded symmetric random walks on any two dimensional discrete lattice.

In our original paper, we provide an expression for the variance of the counting functions associated with the spatial particle configurations formed by infinite systems of independent symmetric alpha-stable processes. The formula (2.3) of the original paper, is in fact the correct expression for the expected conditional number variance. This is equal to the full variance when L is a positive integer multiple of the parameter a but, in general, the full variance has an additional bounded fluctuating term. The main results of the paper still hold for the full variance itself, although some of the proofs require modification in order to incorporate this change.

In [Schuhmacher, Electron. J. Probab. 10 (2005), 165--201] estimates of the Barbour-Brown distance $d_2$ between the distribution of a thinned point process and the distribution of a Poisson process were derived by combining discretization with a result based on Stein's method. In the present article we concentrate on point processes that have a density with respect to a Poisson process, for which we can apply a corresponding result directly without the detour of discretization. This enables us to obtain better and more natural bounds in the $d_2$-metric, and for the first time also bounds in the stronger total variation metric. We give applications for thinning by covering with an independent Boolean model and "Matern type I" thinning of fairly general point processes. These applications give new insight into the respective models, and either generalize or improve earlier results.

A continuous mass population model with local competition is constructed where every emigrant colonizes an unpopulated island. The population founded by an emigrant is modeled as excursion from zero of an one-dimensional diffusion. With this excursion measure, we construct a process which we call Virgin Island Model. A necessary and sufficient condition for extinction of the total population is obtained for finite initial total mass.

We obtain moment and Gaussian bounds for general coordinate-wise Lipschitz functions evaluated along the sample path of a Markov chain. We treat Markov chains on general (possibly unbounded) state spaces via a coupling method. If the first moment of the coupling time exists, then we obtain a variance inequality. If a moment of order $1+a$ $(a > 0)$ of the coupling time exists, then depending on the behavior of the stationary distribution, we obtain higher moment bounds. This immediately implies polynomial concentration inequalities. In the case that a moment of order $1+ a$ is finite, uniformly in the starting point of the coupling, we obtain a Gaussian bound. We illustrate the general results with house of cards processes, in which both uniform and non-uniform behavior of moments of the coupling time can occur.

Let $\{ X, X_i , i \geq 1\}$ be i.i.d. random variables, $S_k$ be the partial sum and $V_n^2 = \sum_{1\leq i\leq n} X_i^2$. Assume that $E(X)=0$ and $E(X^4) < \infty$. In this paper we discuss the moderate deviations of the maximum of the self-normalized sums. In particular, we prove that $P(\max_{1 \leq k \leq n} S_k \geq x V_n) / (1- \Phi(x)) \to 2$ uniformly in $x \in [0, o(n^{1/6}))$.

We derive a precise link between series expansions of Gaussian random vectors in a Banach space and Parseval frames in their reproducing kernel Hilbert space. The results are applied to pathwise continuous Gaussian processes and a new optimal expansion for fractional Ornstein-Uhlenbeck processes is derived.

We define a dynamic model of random networks, where new vertices are connected to old ones with a probability proportional to a sublinear function of their degree. We first give a strong limit law for the empirical degree distribution, and then have a closer look at the temporal evolution of the degrees of individual vertices, which we describe in terms of large and moderate deviation principles. Using these results, we expose an interesting phase transition: in cases of strong preference of large degrees, eventually a single vertex emerges forever as vertex of maximal degree, whereas in cases of weak preference, the vertex of maximal degree is changing infinitely often. Loosely speaking, the transition between the two phases occurs in the case when a new edge is attached to an existing vertex with a probability proportional to the root of its current degree.

We consider an i.i.d. random environment with a strong form of transience on the two dimensional integer lattice. Namely, the walk always moves forward in the y-direction. We prove an invariance principle for the quenched expected position of the random walk indexed by its level crossing times. We begin with a variation of the Martingale Central Limit Theorem. The main part of the paper checks the conditions of the theorem for our problem.

We modify the usual Erdos-Renyi random graph evolution by letting connected clusters 'burn down' (i.e. fall apart to disconnected single sites) due to a Poisson flow of lightnings. In a range of the intensity of rate of lightnings the system sticks to a permanent.

The $(d,\alpha,\beta,\gamma)$-branching particle system consists of particles moving in $\mathbb{R}^d$ according to a symmetric $\alpha$-stable L\'evy process $(0<\alpha\leq 2)$, splitting with a critical $(1+\beta)$-branching law $(0<\beta\leq 1)$, and starting from an inhomogeneous Poisson random measure with intensity measure $\mu_\gamma(dx)=dx/(1+|x|^\gamma), \gamma\geq 0$. By means of time rescaling $T$ and Poisson intensity measure $H_T\mu_\gamma$, occupation time fluctuation limits for the system as $T\to\infty$ have been obtained in two special cases: Lebesgue measure ($\gamma=0$, the homogeneous case), and finite measures $(\gamma > d)$. In some cases $H_T\equiv 1$ and in others $H_T\to\infty$ as $T\to\infty$ (high density systems). The limit processes are quite different for Lebesgue and for finite measures. Therefore the question arises of what kinds of limits can be obtained for Poisson intensity measures that are intermediate between Lebesgue measure and finite measures. In this paper the measures $\mu_\gamma, \gamma\in (0,d]$, are used for investigating this question. Occupation time fluctuation limits are obtained which interpolate in some way between the two previous extreme cases. The limit processes depend on different arrangements of the parameters $d,\alpha,\beta,\gamma$. There are two thresholds for the dimension $d$. The first one, $d=\alpha/\beta+\gamma$, determines the need for high density or not in order to obtain non-trivial limits, and its relation with a.s. local extinction of the system is discussed. The second one, $d=[\alpha(2+\beta)-\gamma\vee \alpha]/\beta$\ (if $\gamma < d$), interpolates between the two extreme cases, and it is a critical dimension which separates different qualitative behaviors of the limit processes, in particular long-range dependence in ``low'' dimensions, and independent increments in ``high'' dimensions. In low dimensions the temporal part of the limit process is a new self-similar stable process which has two different long-range dependence regimes depending on relationships among the parameters. Related results for the corresponding $(d,\alpha,\beta,\gamma)$-superprocess are also given.

We present a quenched weak large deviations principle for the Gibbs measures of a Random Field Kac Model (RFKM) in one dimension. The external random magnetic field is given by symmetrically distributed Bernouilli random variables. The results are valid for values of the temperature and magnitude of the field in the region where the free energy of the corresponding random Curie Weiss model has only two absolute minimizers. We give an explicit representation of the large deviation rate function and characterize its minimizers. We show that they are step functions taking two values, the two absolute minimizers of the free energy of the random Curie Weiss model. The points of discontinuity are described by a stationary renewal process related to the $h$-extrema of a bilateral Brownian motion studied by Neveu and Pitman, where $h$ depends on the temperature and magnitude of the random field. Our result is a complete characterization of the typical profiles of RFKM (the ground states) which was initiated in [2] and extended in [4].

Let $G=(G_{1},G_{2})$ be a Gaussian vector in $R^{2}$ with $E(G_{1}G_{2})\ne 0$. Let $c_{1},c_{2}\in R^{1}$. A necessary and sufficient condition for the vector $((G_{1}+c_{1}\alpha )^{2},(G_{2}+c_{2}\alpha )^{2})$ to be infinitely divisible for all $\alpha \in R^{1}$ is that $$ \Gamma_{i,i}\ge \frac{c_{i}}{c_{j}}\Gamma_{i,j}>0\qquad\forall\,1\le i\ne j\le 2.\qquad(0.1) $$ In this paper we show that when (0.1) does not hold there exists an $0<\alpha _{0} < \infty $ such that $((G_{1}+c_{1}\alpha )^{2},(G_{2}+c_{2}\alpha )^{2})$ is infinitely divisible for all $|\alpha |\le \alpha _{0}$ but not for any $|\alpha | > \alpha _{0}$.

We develop singular value techniques in the context of time inhomogeneous finite Markov chains with the goal of obtaining quantitative results concerning the asymptotic behavior of such chains. We introduce the notion of c-stability which can be viewed as a generalization of the case when a time inhomogeneous chain admits an invariant measure. We describe a number of examples where these techniques yield quantitative results concerning the merging of the distributions of the time inhomogeneous chain started at two arbitrary points.

A functional approach for the study of the random walks in random sceneries (RWRS) is proposed. Under fairly general assumptions on the random walk and on the random scenery, functional limit theorems are proved. The method allows to study separately the convergence of the walk and of the scenery: on the one hand, a general criterion for the convergence of the local time of the walk is provided, on the other hand, the convergence of the random measures associated with the scenery is studied. This functional approach is robust enough to recover many of the known results on RWRS as well as new ones, including the case of many walkers evolving in the same scenery.

We give central lower estimates for the transition kernels corresponding to symmetric random walks on certain amenable p-adic groups.

We construct a martingale which has the same marginals as the arithmetic average of geometric Brownian motion.This provides a short proof of the recent result due to P. Carr et al that the arithmetic average of geometric Brownian motion is increasing in the convex order. The Brownian sheet plays an essential role in the construction. Our method may also be applied when the Brownian motion is replaced by a stable subordinator.

In this paper we study the metastable behavior of one of the simplest disordered spin system, the random field Curie-Weiss model. We will show how the potential theoretic approach can be used to prove sharp estimates on capacities and metastable exit times also in the case when the distribution of the random field is continuous. Previous work was restricted to the case when the random field takes only finitely many values, which allowed the reduction to a finite dimensional problem using lumping techniques. Here we produce the first genuine sharp estimates in a context where entropy is important.

In this article, we first extend the construction of random interlacements, introduced by A.S. Sznitman in [14], to the more general setting of transient weighted graphs. We prove the Harris-FKG inequality for this model and analyze some of its properties on specific classes of graphs. For the case of non-amenable graphs, we prove that the critical value $u_*$ for the percolation of the vacant set is finite. We also prove that, once $\mathcal{G}$ satisfies the isoperimetric inequality $I S_6$ (see (1.5)), $u_*$ is positive for the product $\mathcal{G} \times \mathbb{Z}$ (where we endow $\mathbb{Z}$ with unit weights). When the graph under consideration is a tree, we are able to characterize the vacant cluster containing some fixed point in terms of a Bernoulli independent percolation process. For the specific case of regular trees, we obtain an explicit formula for the critical value $u_*$.

We study a model of multi-excited random walk on a regular tree which generalizes the models of the once excited random walk and the digging random walk introduced by Volkov (2003). We show the existence of a phase transition and provide a criterion for the recurrence/transience property of the walk. In particular, we prove that the asymptotic behaviour of the walk depends on the order of the excitations, which contrasts with the one dimensional setting studied by Zerner (2005). We also consider the limiting speed of the walk in the transient regime and conjecture that it is not a monotonic function of the environment.

We consider simple random walk on a discrete cylinder with base a large $d$-dimensional torus of side-length $N$, when $d$ is two or more. We develop a stochastic domination control on the local picture left by the random walk in boxes of side-length almost of order $N$, at certain random times comparable to the square of the number of sites in the base. We show a domination control in terms of the trace left in similar boxes by random interlacements in the infinite $(d+1)$-dimensional cubic lattice at a suitably adjusted level. As an application we derive a lower bound on the disconnection time of the discrete cylinder, which as a by-product shows the tightness of the laws of the ratio of the square of the number of sites in the base to the disconnection time. This fact had previously only been established when $d$ is at least 17.

In this article, we consider a simple model in equilibrium statistical mechanics for a two-dimensional crystal without defects. In this model, the local specifications for infinite-volume Gibbs measures are rotationally symmetric. We show that at sufficiently low, but positive temperature, rotational symmetry is spontaneously broken in some infinite-volume Gibbs measures.

We consider a randomly perturbed Korteweg-de Vries equation. The perturbation is a random potential depending both on space and time, with a white noise behavior in time, and a regular, but stationary behavior in space. We investigate the dynamics of the soliton of the KdV equation in the presence of this random perturbation, assuming that the amplitude of the perturbation is small. We estimate precisely the exit time of the perturbed solution from a neighborhood of the modulated soliton, and we obtain the modulation equations for the soliton parameters. We moreover prove a central limit theorem for the dispersive part of the solution, and investigate the asymptotic behavior in time of the limit process.

We give three examples of stochastic processes in the Gelfand-Tsetlin cone in which each component evolves independently apart from a blocking and pushing interaction. These processes give rise to couplings between certain conditioned Markov processes, last passage times and exclusion processes. In the first two examples, we deduce known identities in distribution between such processes whilst in the third example, the components of the process cannot escape past a wall at the origin and we obtain a new relation.

We consider Reinforced Random Walks where transitions probabilities are a function of the proportions of times the walk has traversed an edge. We give conditions for recurrence or transience. A phase transition is observed, similar to Pemantle [7] on trees

We present three different fractional versions of the Poisson process and some related results concerning the distribution of order statistics and the compound Poisson process. The main version is constructed by considering the difference-differential equation governing the distribution of the standard Poisson process, $ N(t),t>0$, and by replacing the time-derivative with the fractional Dzerbayshan-Caputo derivative of order $\nu\in(0,1]$. For this process, denoted by $\mathcal{N}_\nu(t),t>0,$ we obtain an interesting probabilistic representation in terms of a composition of the standard Poisson process with a random time, of the form $\mathcal{N}_\nu(t)= N(\mathcal{T}_{2\nu}(t)),$ $t>0$. The time argument $\mathcal{T}_{2\nu }(t),t>0$, is itself a random process whose distribution is related to the fractional diffusion equation. We also construct a planar random motion described by a particle moving at finite velocity and changing direction at times spaced by the fractional Poisson process $\mathcal{N}_\nu.$ For this model we obtain the distributions of the random vector representing the position at time $t$, under the condition of a fixed number of events and in the unconditional case. For some specific values of $\nu\in(0,1]$ we show that the random position has a Brownian behavior (for $\nu =1/2$) or a cylindrical-wave structure (for $\nu =1$).

That there exist two losing games that can be combined, either by random mixture or by nonrandom alternation, to form a winning game is known as Parrondo's paradox. We establish a strong law of large numbers and a central limit theorem for the Parrondo player's sequence of profits, both in a one-parameter family of capital-dependent games and in a two-parameter family of history-dependent games, with the potentially winning game being either a random mixture or a nonrandom pattern of the two losing games. We derive formulas for the mean and variance parameters of the central limit theorem in nearly all such scenarios; formulas for the mean permit an analysis of when the Parrondo effect is present.

The paper studies the filtering problem for a non-classical frame- work: we assume that the observation equation is driven by a signal dependent noise. We show that the support of the conditional distri- bution of the signal is on the corresponding level set of the derivative of the quadratic variation process. Depending on the intrinsic dimension of the noise, we distinguish two cases: In the first case, the conditional distribution has discrete support and we deduce an explicit represen- tation for the conditional distribution. In the second case, the filtering problem is equivalent to a classical one defined on a manifold and we deduce the evolution equation of the conditional distribution. The re- sults are applied to the filtering problem where the observation noise is an Ornstein-Uhlenbeck process.

We prove some new concentration inequalities for self-bounding functions using the entropy method. As an application, we recover Talagrand's convex distance inequality. The new Bernstein-like inequalities for self-bounding functions are derived thanks to a careful analysis of the so-called Herbst argument. The latter involves comparison results between solutions of differential inequalities that may be interesting in their own right.

Using the high moment method and the Feynman-Kac semigroup technique, we obtain moderate deviations and laws of the iterated logarithm for the volume of the intersections of two and three dimensional Wiener sausages.

Consider a vertex-reinforced jump process defined on a regular tree, where each vertex has exactly $b$ children, with $b \geq 3$. We prove the strong law of large numbers and the central limit theorem for the distance of the process from the root. Notice that it is still unknown if vertex-reinforced jump process is transient on the binary tree.

Abstract. For a recurrent linear diffusion on the positive real axis we study the asymptotics of the distribution of its local time at 0 as the time parameter tends to infinity. Under the assumption that the Lévy measure of the inverse local time is subexponential this distribution behaves asymptotically as a multiple of the Lévy measure. Using spectral representations we find the exact value of the multiple. For this we also need a result on the asymptotic behavior of the convolution of a subexponential distribution and an arbitrary distribution on the positive real axis. The exact knowledge of the asymptotic behavior of the distribution of the local time allows us to analyze the process derived via a penalization procedure with the local time. This result generalizes the penalizations obtained by Roynette, Vallois and Yor in Studia Sci. Math. Hungar. 45(1), 2008 for Bessel processes.

We derive general results on the small deviation behavior for some classes of iterated processes. This allows us, in particular, to calculate the rate of the small deviations for n-iterated Brownian motions and, more generally, for the iteration of n fractional Brownian motions. We also give a new and correct proof of some results in E. Nane, Electron. J. Probab. 11 (2006), no. 18, 434--459.

Wiener process with instantaneous reflection in narrow tubes of width $\epsilon\ll 1$ around axis $x$ is considered in this paper. The tube is assumed to be (asymptotically) non-smooth in the following sense. Let $V^{\epsilon}(x)$ be the volume of the cross-section of the tube. We assume that $\frac{1}{\epsilon}V^{\epsilon}(x)$ converges in an appropriate sense to a non-smooth function as $\epsilon\downarrow 0$. This limiting function can be composed by smooth functions, step functions and also the Dirac delta distribution. Under this assumption we prove that the $x$-component of the Wiener process converges weakly to a Markov process that behaves like a standard diffusion process away from the points of discontinuity and has to satisfy certain gluing conditions at the points of discontinuity.

In this paper we consider a model which describes a polymer chain interacting with an infinity of equi-spaced linear interfaces. The distance between two consecutive interfaces is denoted by $T = T_N$ and is allowed to grow with the size $N$ of the polymer. When the polymer receives a positive reward for touching the interfaces, its asymptotic behavior has been derived in Caravenna and Petrelis (2009), showing that a transition occurs when $T_N \approx \log N$. In the present paper, we deal with the so-called depinning case, i.e., the polymer is repelled rather than attracted by the interfaces. Using techniques from renewal theory, we determine the scaling behavior of the model for large $N$ as a function of $\{T_N\}_{N}$, showing that two transitions occur, when $T_N \approx N^{1/3}$ and when $T_N \approx \sqrt{N}$ respectively.

We prove a sharp inequality conjectured by Bobkov on the measure of dilations of Borel sets in the Euclidean space by a $s$-concave probability measure. Our result gives a common generalization of an inequality of Nazarov, Sodin and Volberg and a concentration inequality of Guédon. Applying our inequality to the level sets of functions satisfying a Remez type inequality, we deduce, as it is classical, that these functions enjoy dimension free distribution inequalities and Kahane-Khintchine type inequalities with positive and negative exponent, with respect to an arbitrary $s$-concave probability measure

We continue our study of intermittency for the parabolic Anderson model $\partial u/\partial t = \kappa\Delta u + \xi u$ in a space-time random medium $\xi$, where $\kappa$ is a positive diffusion constant, $\Delta$ is the lattice Laplacian on $\mathbb{Z}^d$, $d \geq 1$, and $\xi$ is a simple symmetric exclusion process on $\mathbb{Z}^d$ in Bernoulli equilibrium. This model describes the evolution of a reactant $u$ under the influence of a catalyst $\xi$. <br /><br /> In Gärtner, den Hollander and Maillard [3] we investigated the behavior of the annealed Lyapunov exponents, i.e., the exponential growth rates as $t\to\infty$ of the successive moments of the solution $u$. This led to an almost complete picture of intermittency as a function of $d$ and $\kappa$. In the present paper we finish our study by focussing on the asymptotics of the Lyaponov exponents as $\kappa\to\infty$ in the critical dimension $d=3$, which was left open in Gärtner, den Hollander and Maillard [3] and which is the most challenging. We show that, interestingly, this asymptotics is characterized not only by a Green term, as in $d\geq 4$, but also by a polaron term. The presence of the latter implies intermittency of all orders above a finite threshold for $\kappa$.

We present a functional central limit theorem for a new class of interacting Markov chain Monte Carlo algorithms. These stochastic algorithms have been recently introduced to solve non-linear measure-valued equations. We provide an original theoretical analysis based on semigroup techniques on distribution spaces and fluctuation theorems for self-interacting random fields. Additionally we also present a series of sharp mean error bounds in terms of the semigroup associated with the first order expansion of the limiting measure-valued process. We illustrate our results in the context of Feynman-Kac semigroups

Consider throwing $n$ balls at random into $m$ urns, each ball landing in urn $i$ with probability $p(i)$. Let $S$ be the resulting number of singletons, i.e., urns containing just one ball. We give an error bound for the Kolmogorov distance from the distribution of $S$ to the normal, and estimates on its variance. These show that if $n$, $m$ and $(p(i))$ vary in such a way that $n p(i)$ remains bounded uniformly in $n$ and $i$, then $S$ satisfies a CLT if and only if ($n$ squared) times the sum of the squares of the entries $p(i)$ tends to infinity, and demonstrate an optimal rate of convergence in the CLT in this case. In the uniform case with all $p(i)$ equal and with $m$ and $n$ growing proportionately, we provide bounds with better asymptotic constants. The proof of the error bounds is based on Stein's method via size-biased couplings.

We prove that a stochastic flow of reflected Brownian motions in a smooth multidimensional domain is differentiable with respect to its initial position. The derivative is a linear map represented by a multiplicative functional for reflected Brownian motion. The method of proof is based on excursion theory and analysis of the deterministic Skorokhod equation.

Almost sure and $L^k$-convergence of the traces of Laguerre processes to the family of dilations of the standard free Poisson distribution are established. We also prove that the fluctuations around the limiting process, converge weakly to a continuous centered Gaussian process. The almost sure convergence on compact time intervals of the largest and smallest eigenvalues processes is also established

We consider the directed first passage percolation model on $\mathbb{Z}^2$. In this model, we assign independently to each edge $e$ a passage time $t(e)$ with a common distribution $F$. We denote by $\vec{T}(0,(r,\theta))$ the passage time from the origin to $(r,\theta)$ by a northeast path for $(r,\theta)\in\mathbb{R}_+\times[0,\pi/2]$. It is known that $\vec{T}(0,(r,\theta))/r$ converges to a time constant $\vec{\mu}_F(\theta)$. Let $\vec{p}_c$ denote the critical probability for oriented percolation. In this paper, we show that the time constant has a phase transition at $\vec{p}_c$, as follows: (1) If $F(0) < \vec{p}_c$, then $\vec{\mu}_F(\theta) > 0$ for all $0 \leq \theta \leq \pi/2$. (2) If $F(0) = \vec{p}_c$, then $\vec{\mu}_F(\theta) > 0$ if and only if $\theta\neq \pi/4$. (3) If $F(0)=p > \vec{p}_c$, then there exists a percolation cone between $\theta_p^-$ and $\theta_p^+$ for $0\leq \theta^-_p < \theta^+_p \leq \pi/2$ such that $\vec{\mu}(\theta) > 0$ if and only if $\theta\not\in[\theta_p^-, \theta^+_p]$. Furthermore, all the moments of $\vec{T}(0, (r, \theta))$ converge whenever $\theta\in[\theta_p^-,\theta^+_p]$. As applications, we describe the shape of the directed growth model on the distribution of $F$. We give a phase transition for the shape at $\vec{p}_c$

We show how to use the Malliavin calculus to obtain a new exact formula for the density $\rho$ of the law of any random variable $Z$ which is measurable and differentiable with respect to a given isonormal Gaussian process. The main advantage of this formula is that it does not refer to the divergence operator $\delta$ (dual of the Malliavin derivative $D$). The formula is based on an auxilliary random variable $G:= < DZ,-DL^{-1}Z >_H$, where $L$ is the generator of the so-called Ornstein-Uhlenbeck semigroup. The use of $G$ was first discovered by Nourdin and Peccati (PTRF 145 75-118 2009 <a href="http://www.ams.org/mathscinet-getitem?mr=2520122">MR-2520122</a>), in the context of rates of convergence in law. Here, thanks to $G$, density lower bounds can be obtained in some instances. Among several examples, we provide an application to the (centered) maximum of a general Gaussian process. We also explain how to derive concentration inequalities for $Z$ in our framework.

We consider the two dimensional discrete Gaussian free field confined between two hard walls. We show that the field becomes massive and identify the precise asymptotic behavior of the mass and the variance of the field as the height of the wall goes to infinity. By large fluctuation of the field, asymptotic behaviors of these quantities in the two dimensional case differ greatly from those of the higher dimensional case studied by [S07].

It is known from the multiplicative ergodic theorem that the norm of the derivative of certain stochastic flows at a previously fixed point grows exponentially fast in time as the flows evolves. We prove that this is also true if one takes the supremum over a bounded set of initial points. We give an explicit bound for the exponential growth rate which is far different from the lower bound coming from the Multiplicative Ergodic Theorem.

The paper establishes a functional version of the Hoeffding combinatorial central limit theorem. First, a pre-limiting Gaussian process approximation is defined, and is shown to be at a distance of the order of the Lyapounov ratio from the original random process. Distance is measured by comparison of expectations of smooth functionals of the processes, and the argument is by way of Stein's method. The pre-limiting process is then shown, under weak conditions, to converge to a Gaussian limit process. The theorem is used to describe the shape of random permutation tableaux.

We study the path behaviour of a simple random walk on the $2$-dimensional comb lattice $C^2$ that is obtained from $\mathbb{Z}^2$ by removing all horizontal edges off the $x$-axis. In particular, we prove a strong approximation result for such a random walk which, in turn, enables us to establish strong limit theorems, like the joint Strassen type law of the iterated logarithm of its two components, as well as their marginal Hirsch type behaviour.

In this paper, we prove that the spectral empirical process of Wigner matrices under sixth-moment conditions, which is indexed by a set of functions with continuous fourth-order derivatives on an open interval including the support of the semicircle law, converges weakly in finite dimensions to a Gaussian process.

We study some remarkable path-properties of generalised stable Fleming-Viot processes (including the so-called spatial Neveu superprocess), inspired by the notion of a "wandering random measure" due to Dawson and Hochberg (1982). In particular, we make use of Donnelly and Kurtz' (1999) modified lookdown construction to analyse their longterm scaling properties, exhibiting a rare natural example of a scaling family of probability laws converging in f.d.d. sense, but not weakly w.r.t. any of Skorohod's topologies on path space. This phenomenon can be explicitly described and intuitively understood in terms of "sparks", leading to the concept of a "flickering random measure". In particular, this completes results of Fleischmann and Wachtel (2006) about the spatial Neveu process and complements results of Dawson and Hochberg (1982) about the classical Fleming Viot process.

This paper deals with stationary Gibbsian point processes on the plane with an interaction that depends on the tiles of the Delaunay triangulation of points via a bounded triangle potential. It is shown that the class of these Gibbs processes includes all minimisers of the associated free energy density and is therefore nonempty. Conversely, each such Gibbs process minimises the free energy density, provided the potential satisfies a weak long-range assumption.

Limiting spectral distribution (LSD) of scaled eigenvalues of circulant, symmetric circulant and a class of k-circulant matrices are known when the input sequence is independent and identically distributed with finite moments of suitable order. We derive the LSD of these matrices when the input sequence is a stationary, two sided moving average process of infinite order. The limits are suitable mixtures of normal, symmetric square root of the chisquare, and other mixture distributions, with the spectral density of the process involved in the mixtures.

We study the asymptotic behavior of the least squares estimators of the unknown parameters of general pth-order bifurcating autoregressive processes. Under very weak assumptions on the driven noise of the process, namely conditional pair-wise independence and suitable moment conditions, we establish the almost sure convergence of our estimators together with the quadratic strong law and the central limit theorem. All our analysis relies on non-standard asymptotic results for martingales.

In this paper we rigorously derive stochastic amplitude equations for a rather general class of SPDEs with quadratic nonlinearities forced by small additive noise. Near a change of stability we use the natural separation of time-scales to show that the solution of the original SPDE is approximated by the solution of an amplitude equation, which describes the evolution of dominant modes. Our results significantly improve older results. We focus on equations with quadratic nonlinearities and give applications to the one-dimensional Burgers’ equation and a model from surface growth.

LDP is proved for the inviscid shell model of turbulence. As the viscosity coefficient converges to 0 and the noise intensity is multiplied by its square root, we prove that some shell models of turbulence with a multiplicative stochastic perturbation driven by a $H$-valued Brownian motion satisfy a LDP in $\mathcal{C}([0,T],V)$ for the topology of uniform convergence on $[0,T]$, but where $V$ is endowed with a topology weaker than the natural one. The initial condition has to belong to $V$ and the proof is based on the weak convergence of a family of stochastic control equations. The rate function is described in terms of the solution to the inviscid equation.

We consider reversible random walks in random environment obtained from symmetric long-range jump rates on a random point process. We prove almost sure transience and recurrence results under suitable assumptions on the point process and the jump rate function. For recurrent models we obtain almost sure estimates on effective resistances in finite boxes. For transient models we construct explicit fluxes with finite energy on the associated electrical network.

Let $X_1,\ldots,X_n$ be $n$ independent observations drawn from a multivariate probability density $f$ with compact support $S_f$. This paper is devoted to the study of the estimator $\hat{S}_n$ of $S_f$ defined as the union of balls centered at the $X_i$ and with common radius $r_n$. Using tools from Riemannian geometry, and under mild assumptions on $f$ and the sequence $(r_n)$, we prove a central limit theorem for $\lambda (S_n \Delta S_f)$, where $\lambda$ denotes the Lebesgue measure on $\mathbb{R}^d$ and $\Delta$ the symmetric difference operation.

We prove the moderate deviation principle for subgraph count statistics of Erdös-Rényi random graphs. This is equivalent in showing the moderate deviation principle for the trace of a power of a Bernoulli random matrix. It is done via an estimation of the log-Laplace transform and the Gärtner-Ellis theorem. We obtain upper bounds on the upper tail probabilities of the number of occurrences of small subgraphs. The method of proof is used to show supplemental moderate deviation principles for a class of symmetric statistics, including non-degenerate U-statistics with independent or Markovian entries.

For Brownian motion in an unbounded domain we study the influence of the "far away" behavior of the domain on the probability that the modulus of the Brownian motion is large when it exits the domain. Roughly speaking, if the domain expands at a sublinear rate, then the chance of a large exit place decays in a subexponential fashion. The decay rate can be explicitly given in terms of the sublinear expansion rate. Our results encompass and extend some known special cases.

The aim of the present paper is to investigate series representations of the Riemann-Liouville process $R^\alpha$, $\alpha >1/2$, generated by classical orthonormal bases in $L_2[0,1]$. Those bases are, for example, the trigonometric or the Haar system. We prove that the representation of $R^\alpha$ via the trigonometric system possesses the optimal convergence rate if and only if $1/2 < \alpha\leq 2$. For the Haar system we have an optimal approximation rate if $1/2 < \alpha <3/2$ while for $\alpha > 3/2$ a representation via the Haar system is not optimal. Estimates for the rate of convergence of the Haar series are given in the cases $\alpha > 3/2$ and $\alpha = 3/2$. However, in this latter case the question whether or not the series representation is optimal remains open. Recently M. A. Lifshits answered this question (cf. [13]). Using a different approach he could show that in the case $\alpha = 3/2$ a representation of the Riemann-Liouville process via the Haar system is also not optimal.