For a symmetric random walk in $Z^2$ which does not necessarily have bounded jumps we study those points which are visited an unusually large number of times. We prove the analogue of the Erdös-Taylor conjecture and obtain the asymptotics for the number of visits to the most visited site. We also obtain the asymptotics for the number of points which are visited very frequently by time $n$. Among the tools we use are Harnack inequalities and Green's function estimates for random walks with unbounded jumps; some of these are of independent interest.

We resolve the longstanding question of how to define the compensator of a point process on a general partially ordered set in such a way that the compensator exists, is unique, and characterizes the law of the process. We define a family of one-parameter compensators and prove that this family is unique in some sense and characterizes the finite dimensional distributions of a totally ordered point process. This result can then be applied to a general point process since we prove that such a process can be embedded into a totally ordered point process on a larger space. We present some examples, including the partial sum multiparameter process, single line point processes, multiparameter renewal processes, and obtain a new characterization of the two-parameter Poisson process

In the supermarket model there are $n$ queues, each with a unit rate server. Customers arrive in a Poisson process at rate $\lambda n$, where $0<\lambda<1$. Each customer chooses $d \geq 2$ queues uniformly at random, and joins a shortest one. It is known that the equilibrium distribution of a typical queue length converges to a certain explicit limiting distribution as $n\to\infty$. We quantify the rate of convergence by showing that the total variation distance between the equilibrium distribution and the limiting distribution is essentially of order $1/n$ and we give a corresponding result for systems starting from quite general initial conditions (not in equilibrium). Further, we quantify the result that the systems exhibit chaotic behaviour: we show that the total variation distance between the joint law of a fixed set of queue lengths and the corresponding product law is essentially of order at most $1/n$.

We provide several inequalities on the asymptotic behavior of the harmonic measure of the first exit position of a $d$-dimensional symmetric stable process from a unbounded convex domain. Our results on the harmonic measure will determine the asymptotic behavior of the distributions of the first exit time from the domain. These inequalities are given in terms of the growth of the inradius of the cross sections of the domain.

Our first aim in this paper is to characterize Palm measures of stationary point processes through point stationarity. This generalizes earlier results from the Euclidean case to the case of an Abelian group. While a stationary point process looks statistically the same from each site, a point stationary point process looks statistically the same from each of its points. Even in the Euclidean case our proof will simplify some of the earlier arguments. A new technical result of some independent interest is the existence of a complete countable family of matchings. Using a change of measure we will generalize our results to discrete random measures. In the Euclidean case we will finally treat general random measures by means of a suitable approximation.

In this paper we shall derive asymptotic expansions of the Green function and the transition probabilities of Markov additive (MA) processes $(\xi_n, S_n)$ whose first component satisfies Doeblin's condition and the second one takes valued in $Z^d$. The derivation is based on a certain perturbation argument that has been used in previous works in the same context. In our asymptotic expansions, however, not only the principal term but also the second order term are expressed explicitly in terms of a few basic functions that are characteristics of the expansion. The second order term will be important for instance in computation of the harmonic measures of a half space for certain models. We introduce a certain aperiodicity condition, named Condition (AP), that seems a minimal one under which the Fourier analysis can be applied straightforwardly. In the case when Condition (AP) is violated the structure of MA processes will be clarified and it will be shown that in a simple manner the process, if not degenerate, are transformed to another one that satisfies Condition (AP) so that from it we derive either directly or indirectly (depending on purpose) the asymptotic expansions for the original process. It in particular is shown that if the MA processes is irreducible as a Markov process, then the Green function is expanded quite similarly to that of a classical random walk on $Z^d$.

Stable non-Gaussian self-similar mixed moving averages can be decomposed into several components. Two of these are the periodic and cyclic fractional stable motions which are the subject of this study. We focus on the structure of their integral representations and show that the periodic fractional stable motions have, in fact, a canonical representation. We study several examples and discuss questions of uniqueness, namely how to determine whether two given integral representations of periodic or cyclic fractional stable motions give rise to the same process.

Under the hypothesis of convergence in probability of a sequence of càdlàg processes $(X^n)$ to a càdlàg process $X$, we are interested in the convergence of corresponding values in optimal stopping and also in the convergence of optimal stopping times. We give results under hypothesis of inclusion of filtrations or convergence of filtrations.

A Bernoulli random walk is a random trajectory starting from 0 and having i.i.d. increments, each of them being +1 or -1, equally likely. The other families quoted in the title are Bernoulli random walks under various conditions. A peak in a trajectory is a local maximum. In this paper, we condition the families of trajectories to have a given number of peaks. We show that, asymptotically, the main effect of setting the number of peaks is to change the order of magnitude of the trajectories. The counting process of the peaks, that encodes the repartition of the peaks in the trajectories, is also studied. It is shown that suitably normalized, it converges to a Brownian bridge which is independent of the limiting trajectory. Applications in terms of plane trees and parallelogram polyominoes are provided, as well as an application to the ``comparison'' between runs and Kolmogorov-Smirnov statistics.

In this paper, we develop a new "robust mixing" framework for reasoning about adversarially modified Markov Chains (AMMC). Let $\mathbb{P}$ be the transition matrix of an irreducible Markov Chain with stationary distribution $\pi$. An adversary announces a sequence of stochastic matrices $\{\mathbb{A}_t\}_{t > 0}$ satisfying $\pi\mathbb{A}_t = \pi$. An AMMC process involves an application of $\mathbb{P}$ followed by $\mathbb{A}_t$ at time $t$. The robust mixing time of an ergodic Markov Chain $\mathbb{P}$ is the supremum over all adversarial strategies of the mixing time of the corresponding AMMC process. Applications include estimating the mixing times for certain non-Markovian processes and for reversible liftings of Markov Chains.

Non-Markovian card shuffling processes: The random-to-cyclic transposition process is a non-Markovian card shuffling process, which at time $t$, exchanges the card at position $L_t := t {\pmod n}$ with a random card. Mossel, Peres and Sinclair (2004) showed a lower bound of $(0.0345+o(1))n\log n$ for the mixing time of the random-to-cyclic transposition process. They also considered a generalization of this process where the choice of $L_t$ is adversarial, and proved an upper bound of $C n\log n + O(n)$ (with $C \approx 4\times 10^5$) on the mixing time. We reduce the constant to $1$ by showing that the random-to-top transposition chain (a Markov Chain) has robust mixing time $\leq n\log n + O(n)$ when the adversarial strategies are limited to holomorphic strategies, i.e. those strategies which preserve the symmetry of the underlying Markov Chain. We also show a $O(n\log^2 n)$ bound on the robust mixing time of the lazy random-to-top transposition chain when the adversary is not limited to holomorphic strategies.

Reversible liftings: Chen, Lovasz and Pak showed that for a reversible ergodic Markov Chain $\mathbb{P}$, any reversible lifting $\mathbb{Q}$ of $\mathbb{P}$ must satisfy $\mathcal{T}(\mathbb{P}) \leq \mathcal{T}(\mathbb{Q})\log (1/\pi_*)$ where $\pi_*$ is the minimum stationary probability. Looking at a specific adversarial strategy allows us to show that $\mathcal{T}(\mathbb{Q}) \geq r(\mathbb{P})$ where $r(\mathbb{P})$ is the relaxation time of $\mathbb{P}$. This gives an alternate proof of the reversible lifting result and helps identify cases where reversible liftings cannot improve the mixing time by more than a constant factor.

Consider the high-order heat-type equation $\partial u/\partial t=\pm\partial^N u/\partial x^N$ for an integer $N>2$ and introduce the related Markov pseudo-process $(X(t))_{t\ge 0}$. In this paper, we study several functionals related to $(X(t))_{t\ge 0}$: the maximum $M(t)$ and minimum $m(t)$ up to time $t$; the hitting times $\tau_a^+$ and $\tau_a^-$ of the half lines $(a,+\infty)$ and $(-\infty,a)$ respectively. We provide explicit expressions for the distributions of the vectors $(X(t),M(t))$ and $(X(t),m(t))$, as well as those of the vectors $(\tau_a^+,X(\tau_a^+))$ and $(\tau_a^-,X(\tau_a^-))$.

We investigate the evolution of the random interfaces in a two dimensional Potts model at zero temperature under Glauber dynamics for some particular initial conditions. We prove that under space-time diffusive scaling the shape of the interfaces converges in probability to the solution of a non-linear parabolic equation. This Law of Large Numbers is obtained from the Hydrodynamic limit of a coupling between an exclusion process and an inhomogeneous one dimensional zero range process with asymmetry at the origin.

We study regularizing properties for transition semigroups related to Ornstein Uhlenbeck processes with values in a Banach space $E$ which is continuously and densely embedded in a real and separable Hilbert space $H$. Namely we study conditions under which the transition semigroup maps continuous and bounded functions into differentiable functions. Via a Girsanov type theorem such properties extend to perturbed Ornstein Uhlenbeck processes. We apply the results to solve in mild sense semilinear versions of Kolmogorov equations in $E$.

We consider continuous-state branching (CB) processes which become extinct (i.e., hit 0) with positive probability. We characterize all the quasi-stationary distributions (QSD) for the CB-process as a stochastically monotone family indexed by a real number. We prove that the minimal element of this family is the so-called Yaglom quasi-stationary distribution, that is, the limit of one-dimensional marginals conditioned on being nonzero. Next, we consider the branching process conditioned on not being extinct in the distant future, or $Q$-process, defined by means of Doob $h$-transforms. We show that the $Q$-process is distributed as the initial CB-process with independent immigration, and that under the $L\log L$ condition, it has a limiting law which is the size-biased Yaglom distribution (of the CB-process). More generally, we prove that for a wide class of nonnegative Markov processes absorbed at 0 with probability 1, the Yaglom distribution is always stochastically dominated by the stationary probability of the $Q$-process, assuming that both exist. Finally, in the diffusion case and in the stable case, the $Q$-process solves a SDE with a drift term that can be seen as the instantaneous immigration.

We consider generalized adapted stochastic integrals with respect to independently scattered random measures with second moments, and use a decoupling technique, formulated as a «principle of conditioning», to study their stable convergence towards mixtures of infinitely divisible distributions. The goal of this paper is to develop the theory. Our results apply, in particular, to Skorohod integrals on abstract Wiener spaces, and to multiple integrals with respect to independently scattered and finite variance random measures. The first application is discussed in some detail in the final sectionof the present work, and further extended in a companion paper (Peccati and Taqqu (2006b)). Applications to the stable convergence (in particular, central limit theorems) of multiple Wiener-Itô integrals with respect to independently scattered (and not necessarily Gaussian) random measures are developed in Peccati and Taqqu (2006a, 2007). The present work concludes with an example involving quadratic Brownian functionals.

We show that if $G=(V,E)$ is a regular bipartite graph for which the expansion of subsets of a single parity of $V$ is reasonably good and which satisfies a certain local condition (that the union of the neighbourhoods of adjacent vertices does not contain too many pairwise non-adjacent vertices), and if $M$ is a Markov chain on the set of proper 3-colourings of $G$ which updates the colour of at most $c|V|$ vertices at each step and whose stationary distribution is uniform, then for $c < .22$ and $d$ sufficiently large the convergence to stationarity of $M$ is (essentially) exponential in $|V|$. In particular, if $G$ is the $d$-dimensional hypercube $Q_d$ (the graph on vertex set $\{0,1\}^d$ in which two strings are adjacent if they differ on exactly one coordinate) then the convergence to stationarity of the well-known Glauber (single-site update) dynamics is exponentially slow in $2^d/(\sqrt{d} \log d )$. A combinatorial corollary of our main result is that in a uniform 3-colouring of $Q_d$ there is an exponentially small probability (in $2^d$) that there is a colour $i$ such the proportion of vertices of the even subcube coloured $i$ differs from the proportion of the odd subcube coloured $i$ by at most .22. Our proof combines a conductance argument with combinatorial enumeration methods.

We investigate a possible definition of expectation and conditional expectation for random variables with values in a local field such as the $p$-adic numbers. We define the expectation by analogy with the observation that for real-valued random variables in $L^2$ the expected value is the orthogonal projection onto the constants. Previous work has shown that the local field version of $L^\infty$ is the appropriate counterpart of $L^2$, and so the expected value of a local field-valued random variable is defined to be its ``projection'' in $L^\infty$ onto the constants. Unlike the real case, the resulting projection is not typically a single constant, but rather a ball in the metric on the local field. However, many properties of this expectation operation and the corresponding conditional expectation mirror those familiar from the real-valued case; for example, conditional expectation is, in a suitable sense, a contraction on $L^\infty$ and the tower property holds. We also define the corresponding notion of martingale, show that several standard examples of martingales (for example, sums or products of suitable independent random variables or ``harmonic'' functions composed with Markov chains) have local field analogues, and obtain versions of the optional sampling and martingale convergence theorems.

We continue our study of intermittency for the parabolic Anderson equation, i.e., the spatially discrete heat equation on the d-dimensional integer lattice with a space-time random potential. The solution of the equation describes the evolution of a "reactant" under the influence of a "catalyst". <br> In this paper we focus on the case where the random field is an exclusion process with a symmetric random walk transition kernel, starting from Bernoulli equilibrium. We consider the annealed Lyapunov exponents, i.e., the exponential growth rates of the successive moments of the solution. We show that these exponents are trivial when the random walk is recurrent, but display an interesting dependence on the diffusion constant when the random walk is transient, with qualitatively different behavior in different dimensions. Special attention is given to the asymptotics of the exponents when the diffusion constant tends to infinity, which is controlled by moderate deviations of the random field requiring a delicate expansion argument. <br> In G&#228;rtner and den Hollander [10] the case of a Poisson field of independent (simple) random walks was studied. The two cases show interesting differences and similarities. Throughout the paper, a comparison of the two cases plays a crucial role.

A reflected Brownian motion in the Gelfand-Tsetlin cone is used to construct Dyson's process of non-colliding Brownian motions. The key step of the construction is to consider two interlaced families of Brownian paths with paths belonging to the second family reflected off paths belonging to the first. Such families of paths are known to arise in the Arratia flow of coalescing Brownian motions. A determinantal formula for the distribution of coalescing Brownian motions is presented.

Let $G$ be a vertex transitive graph. A study of the range of simple random walk on $G$ and of its bridge is proposed. While it is expected that on a graph of polynomial growth the sizes of the range of the unrestricted random walk and of its bridge are the same in first order, this is not the case on some larger graphs such as regular trees. Of particular interest is the case when $G$ is the Cayley graph of a group. In this case we even study the range of a general symmetric (not necessarily simple) random walk on $G$. We hope that the few examples for which we calculate the first order behavior of the range here will help to discover some relation between the group structure and the behavior of the range. Further problems regarding bridges are presented.

We consider models of directed polymers interacting with a one-dimensional defect line on which random charges are placed. More abstractly, one starts from renewal sequence on $Z$ and gives a random (site-dependent) reward or penalty to the occurrence of a renewal at any given point of $Z$. These models are known to undergo a delocalization-localization transition, and the free energy $F$ vanishes when the critical point is approached from the localized region. We prove that the quenched correlation length $\xi$, defined as the inverse of the rate of exponential decay of the two-point function, does not diverge faster than $1/F$. We prove also an exponentially decaying upper bound for the disorder-averaged two-point function, with a good control of the sub-exponential prefactor. We discuss how, in the particular case where disorder is absent, this result can be seen as a refinement of the classical renewal theorem, for a specific class of renewal sequences.

A continuous one-dimensional scenery is a double-infinite sequence of points (thought of as locations of <em>bells</em>) in $R$. Assume that a scenery $X$ is observed along the path of a Brownian motion in the following way: when the Brownian motion encounters a bell different from the last one visited, we hear a ring. The trajectory of the Brownian motion is unknown, whilst the scenery $X$ is known except in some finite interval. We prove that given only the sequence of times of rings, we can a.s. reconstruct the scenery $X$ entirely. For this we take the scenery$X$ to be a local perturbation of a Poisson scenery $X'$. We present an explicit reconstruction algorithm. This problem is the continuous analog of the "detection of a defect in a discrete scenery". Many of the essential techniques used with discrete sceneries do not work with continuous sceneries.

We consider finite-state time-nonhomogeneous Markov chains whose transition matrix at time $n$ is $I+G/n^z$ where $G$ is a ``generator'' matrix, that is $G(i,j)>0$ for $i,j$ distinct, and $G(i,i)= -\sum_{k\ne i} G(i,k)$, and $z>0$ is a strength parameter. In these chains, as time grows, the positions are less and less likely to change, and so form simple models of age-dependent time-reinforcing schemes. These chains, however, exhibit a trichotomy of occupation behaviors depending on parameters.

We show that the average occupation or empirical distribution vector up to time $n$, when variously $0< z< 1$, $z>1$ or $z=1$, converges in probability to a unique ``stationary'' vector $n_G$, converges in law to a nontrivial mixture of point measures, or converges in law to a distribution $m_G$ with no atoms and full support on a simplex respectively, as $n$ tends to infinity. This last type of limit can be interpreted as a sort of ``spreading'' between the cases $0< z < 1$ and $z>1$.

In particular, when $G$ is appropriately chosen, $m_G$ is a Dirichlet distribution, reminiscent of results in Polya urns.

We consider an irreducible pure jump Markov process with rates $Q=(q(x,y))$ on $\Lambda\cup\{0\}$ with $\Lambda$ countable and $0$ an absorbing state. A {\em quasi stationary distribution \rm} (QSD) is a probability measure $\nu$ on $\Lambda$ that satisfies: starting with $\nu$, the conditional distribution at time $t$, given that at time $t$ the process has not been absorbed, is still $\nu$. That is, $\nu(x) = \nu P_t(x)/(\sum_{y\in\Lambda}\nu P_t(y))$, with $P_t$ the transition probabilities for the process with rates $Q$.

A Fleming-Viot (FV) process is a system of $N$ particles moving in $\Lambda$. Each particle moves independently with rates $Q$ until it hits the absorbing state $0$; but then instantaneously chooses one of the $N-1$ particles remaining in $\Lambda$ and jumps to its position. Between absorptions each particle moves with rates $Q$ independently.

Under the condition $\alpha:=\sum_{x\in\Lambda}\inf Q(\cdot,x) > \sup Q(\cdot,0):=C$ we prove existence of QSD for $Q$; uniqueness has been proven by Jacka and Roberts. When $\alpha>0$ the FV process is ergodic for each $N$. Under $\alpha>C$ the mean normalized densities of the FV unique stationary measure converge to the QSD of $Q$, as $N \to \infty$; in this limit the variances vanish.

In this paper we study typical distances in random graphs with i.i.d. degrees of which the tail of the common distribution function is regularly varying with exponent $1-\tau$. Depending on the value of the parameter $\tau$ we can distinct three cases: (i) $\tau>3$, where the degrees have finite variance, (ii) $\tau\in (2,3)$, where the degrees have infinite variance, but finite mean, and (iii) $\tau\in (1,2)$, where the degrees have infinite mean. The distances between two randomly chosen nodes belonging to the same connected component, for $\tau>3$ and $\tau \in (1,2),$ have been studied in previous publications, and we survey these results here. When $\tau\in (2,3)$, the graph distance centers around $2\log\log{N}/|\log(\tau-2)|$. We present a full proof of this result, and study the fluctuations around this asymptotic means, by describing the asymptotic distribution. The results presented here improve upon results of Reittu and Norros, who prove an upper bound only.

The random graphs studied here can serve as models for complex networks where degree power laws are observed; this is illustrated by comparing the typical distance in this model to Internet data, where a degree power law with exponent $\tau\approx 2.2$ is observed for the so-called Autonomous Systems (AS) graph

Chain records is a new type of multidimensional record. We discuss how often the chain records occur when the background sampling is from the unit cube with uniform distribution (or, more generally, from an arbitrary continuous product distribution in d dimensions). Extensions are given for sampling from more general spaces with a self-similarity property.

Large deviation principles are established for the two-parameter Poisson-Dirichlet distribution and two-parameter Dirichlet process when parameter $\theta$ approaches infinity. The motivation for these results is to understand the differences in terms of large deviations between the two-parameter models and their one-parameter counterparts. New insight is obtained about the role of the second parameter $\alpha$ through a comparison with the corresponding results for the one-parameter Poisson-Dirichlet distribution and Dirichlet process.

Rates of molecular evolution along phylogenetic trees are influenced by mutation, selection and genetic drift. Provided that the branches of the tree correspond to lineages belonging to genetically isolated populations (e.g., multi-species phylogenies), the interplay between these three processes can be described by analyzing the process of substitutions to the common ancestor of each population. We characterize this process for a class of diffusion models from population genetics theory using the structured coalescent process introduced by Kaplan et al. (1988) and formalized in Barton et al. (2004). For two-allele models, this approach allows both the stationary distribution of the type of the common ancestor and the generator of the common ancestor process to be determined by solving a one-dimensional boundary value problem. In the case of a Wright-Fisher diffusion with genic selection, this solution can be found in closed form, and we show that our results complement those obtained by Fearnhead (2002) using the ancestral selection graph. We also observe that approximations which neglect recurrent mutation can significantly underestimate the exact substitution rates when selection is strong. Furthermore, although we are unable to find closed-form expressions for models with frequency-dependent selection, we can still solve the corresponding boundary value problem numerically and then use this solution to calculate the substitution rates to the common ancestor. We illustrate this approach by studying the effect of dominance on the common ancestor process in a diploid population. Finally, we show that the theory can be formally extended to diffusion models with more than two genetic backgrounds, but that it leads to systems of singular partial differential equations which we have been unable to solve.

We consider stochastic nonlinear Schrodinger equations driven by an additive noise. The noise is fractional in time with Hurst parameter $H \in (0,1)$ and colored in space with a nuclear space correlation operator. We study local well-posedness. Under adequate assumptions on the initial data, the space correlations of the noise and for some saturated nonlinearities, we prove sample path large deviations and support results in a space of Holder continuous in time until blow-up paths. We consider Kerr nonlinearities when $H > 1/2$.

Some probabilistic aspects of the number variance statistic are investigated. Infinite systems of independent Brownian motions and symmetric alpha-stable processes are used to construct explicit new examples of processes which exhibit both divergent and saturating number variance behaviour. We derive a general expression for the number variance for the spatial particle configurations arising from these systems and this enables us to deduce various limiting distribution results for the fluctuations of the associated counting functions. In particular, knowledge of the number variance allows us to introduce and characterize a novel family of centered, long memory Gaussian processes. We obtain fractional Brownian motion as a weak limit of these constructed processes.

We prove some asymptotic results for the radius and the profile of large random bipartite planar maps. Using a bijection due to Bouttier, Di Francesco and Guitter between rooted bipartite planar maps and certain two-type trees with positive labels, we derive our results from a conditional limit theorem for two-type spatial trees. Finally we apply our estimates to separating vertices of bipartite planar maps: with probability close to one when n tends to infinity, a random $2k$-angulation with n faces has a separating vertex whose removal disconnects the map into two components each with size greater that $n^{1/2 - \varepsilon}$.

A graph homomorphism between two graphs is a map from the vertex set of one graph to the vertex set of the other graph, that maps edges to edges. In this note we study the range of a uniformly chosen homomorphism from a graph $G$ to the infinite line $Z$. It is shown that if the maximal degree of $G$ is `sub-logarithmic', then the range of such a homomorphism is super-constant.

Furthermore, some examples are provided, suggesting that perhaps for graphs with super-logarithmic degree, the range of a typical homomorphism is bounded. In particular, a sharp transition is shown for a specific family of graphs $C_{n,k}$ (which is the tensor product of the $n$-cycle and a complete graph, with self-loops, of size $k$). That is, given any function $\psi(n)$ tending to infinity, the range of a typical homomorphism of $C_{n,k}$ is super-constant for $k= 2\log(n) - \psi(n)$, and is $3$ for $k= 2\log(n) + \psi(n)$.

A general version of the Yamada-Watanabe and Engelbert results relating existence and uniqueness of strong and weak solutions for stochastic equations is given. The results apply to a wide variety of stochastic equations including classical stochastic differential equations, stochastic partial differential equations, and equations involving multiple time transformations.

We prove such a multivariate version of Bernstein's inequality about the tail distribution of degenerate $U$-statistics which is an improvement of some former results. This estimate will be compared with an analogous bound about the tail distribution of multiple Wiener-Ito integrals. Their comparison shows that our estimate is sharp. The proof is based on good estimates about high moments of degenerate $U$-statistics. They are obtained by means of a diagram formula which enables us to express the product of degenerate $U$-statistics as the sum of such expressions.

Given $n$ independent random marked $d$-vectors $X_i$ with a common density, define the measure $\nu_n = \sum_i \xi_i $, where $\xi_i$ is a measure (not necessarily a point measure) determined by the (suitably rescaled) set of points near $X_i$. Technically, this means here that $\xi_i$ stabilizes with a suitable power-law decay of the tail of the radius of stabilization. For bounded test functions $f$ on $R^d$, we give a central limit theorem for $\nu_n(f)$, and deduce weak convergence of $\nu_n(\cdot)$, suitably scaled and centred, to a Gaussian field acting on bounded test functions. The general result is illustrated with applications to measures associated with germ-grain models, random and cooperative sequential adsorption, Voronoi tessellation and $k$-nearest neighbours graph.

We consider various models of randomly grown graphs. In these models the vertices and the edges accumulate within time according to certain rules. We study a phase transition in these models along a parameter which refers to the mean life-time of an edge. Although deleting old edges in the uniformly grown graph changes abruptly the properties of the model, we show that some of the macro-characteristics of the graph vary continuously. In particular, our results yield a lower bound for the size of the largest connected component of the uniformly grown graph.

In the paper Johansson and Nordenstam (2006) [1] two expressions for the so called GUE minor kernel are presented, one in definition 1.2 and one in the the formulas (5.6) and (5.7). The expressions given in (5.6) and (5.7) are correct, but the expression in definition 1.2 of the paper has to be modified in the case $r > s$. The proof of the equality of the two expressions for the GUE minor kernel given in the paper was based on lemma 5.6 which is not correct since some terms in the expansion are missing. The correct expansion is given in lemma 1.2 below.

Let $\{(Z_i,W_i):i=1,\dots,n\}$ be uniformly distributed in $[0,1]^d \times \mathbb{G}(k,d)$, where $\mathbb{G}(k,d)$ denotes the space of $k$-dimensional linear subspaces of $\mathbb{R}^d$. For a differentiable function $f: [0,1]^k \rightarrow [0,1]^d$, we say that $f$ interpolates $(z,w) \in [0,1]^d \times \mathbb{G}(k,d)$ if there exists $x \in [0,1]^k$ such that $f(x) = z$ and $\vec{f}(x) = w$, where $\vec{f}(x)$ denotes the tangent space at $x$ defined by $f$. For a smoothness class ${\cal F}$ of Holder type, we obtain probability bounds on the maximum number of points a function $f \in {\cal F}$ interpolates.

Large deviations and isoperimetric inequalities are considered for probability distributions, satisfying convexity conditions of the Brunn-Minkowski-type

The frequencies of an exchangeable Gibbs random partition of the integers (Gnedin and Pitman 2005) are considered in their age-order, i.e. their size-biased order. We study their dependence on the sequence of record indices (i.e. the least elements) of the blocks of the partition. In particular we show that, conditionally on the record indices, the distribution of the age-ordered frequencies has a left-neutral stick-breaking structure. Such a property in fact characterizes the Gibbs family among all exchangeable partitions and leads to further interesting results on: (i) the conditional Mellin transform of the $k$-th oldest frequency given the $k$-th record index, and (ii) the conditional distribution of the first $k$ normalized frequencies, given their sum and the $k$-th record index; the latter turns out to be a mixture of Dirichlet distributions. Many of the mentioned representations are extensions of Griffiths and Lessard (2005) results on Ewens' partitions.

We establish a large deviation principle for the largest eigenvalue of a rank one deformation of a matrix from the GUE or GOE. As a corollary, we get another proof of the phenomenon, well-known in learning theory and finance, that the largest eigenvalue separates from the bulk when the perturbation is large enough. A large part of the paper is devoted to an auxiliary result on the continuity of spherical integrals in the case when one of the matrix is of rank one, as studied in one of our previous works.

An acyclic mapping from an $n$ element set into itself is a mapping $\varphi$ such that if $\varphi^k(x) = x$ for some $k$ and $x$, then $\varphi(x) = x$. Equivalently, $\varphi^\ell = \varphi^{\ell+1} = \ldots$ for $\ell$ sufficiently large. We investigate the behavior as $n \rightarrow \infty$ of a sequence of a Markov chain on the collection of such mappings. At each step of the chain, a point in the $n$ element set is chosen uniformly at random and the current mapping is modified by replacing the current image of that point by a new one chosen independently and uniformly at random, conditional on the resulting mapping being again acyclic. We can represent an acyclic mapping as a directed graph (such a graph will be a collection of rooted trees) and think of these directed graphs as metric spaces with some extra structure. Informal calculations indicate that the metric space valued process associated with the Markov chain should, after an appropriate time and ``space'' rescaling, converge as $n \rightarrow \infty$ to a real tree ($R$-tree) valued Markov process that is reversible with respect to a measure induced naturally by the standard reflected Brownian bridge. Although we don't prove such a limit theorem, we use Dirichlet form methods to construct a Markov process that is Hunt with respect to a suitable Gromov-Hausdorff-like metric and evolves according to the dynamics suggested by the heuristic arguments. This process is similar to one that appears in earlier work by Evans and Winter as a similarly informal limit of a Markov chain related to the subtree prune and regraft tree (SPR) rearrangements from phylogenetics.

Let $X$ be a drifted fractional Brownian motion with Hurst index $H > 1/2$. We prove that there exists a fractional backward representation of $X$, i.e. the time reversed process is a drifted fractional Brownian motion, which continuously extends the one obtained in the theory of time reversal of Brownian diffusions when $H=1/2$. We then apply our result to stochastic differential equations driven by a fractional Brownian motion.

We study the isoperimetric problem for product probability measures with tails between the exponential and the Gaussian regime. In particular we exhibit many examples where coordinate half-spaces are approximate solutions of the isoperimetric problem

For the plane, sphere, and hyperbolic plane we consider the canonical invariant determinantal point processes $\mathcal Z_\rho$ with intensity $\rho d\nu$, where $\nu$ is the corresponding invariant measure. We show that as $\rho\to\infty$, after centering, these processes converge to invariant $H^1$ noise. More precisely, for all functions $f\in H^1(\nu) \cap L^1(\nu)$ the distribution of $\sum_{z\in \mathcal Z} f(z)-\frac{\rho}{\pi} \int f d \nu$ converges to Gaussian with mean zero and variance $ \frac{1}{4 \pi} \|f\|_{H^1}^2$.

We introduce a class of rooted infinite self-similar graphs containing the well known Fibonacci graph and graphs associated with Pisot numbers. We consider directed random walks on these graphs and study their entropy and their limit measures. We prove that every infinite self-similar graph has a random walk of full entropy and that the limit measures of this random walks are absolutely continuous.

Let $X_1, X_2, \dots$ be independent and symmetric random variables such that $S_n = X_1 + \cdots + X_n$ converges to a finite valued random variable $S$ a.s. and let $S^* = \sup_{1 \leq n \leq \infty} S_n$ (which is finite a.s.). We construct upper and lower bounds for $s_y$ and $s_y^*$, the upper $1/y$-th quantile of $S_y$ and $S^*$, respectively. Our approximations rely on an explicitly computable quantity $\underline q_y$ for which we prove that $$\frac 1 2 \underline q_{y/2} < s_y^* < 2 \underline q_{2y} \quad \text{ and } \quad \frac 1 2 \underline q_{ (y/4) ( 1 + \sqrt{ 1 - 8/y})} < s_y < 2 \underline q_{2y}. $$ The RHS's hold for $y \geq 2$ and the LHS's for $y \geq 94$ and $y \geq 97$, respectively. Although our results are derived primarily for symmetric random variables, they apply to non-negative variates and extend to an absolute value of a sum of independent but otherwise arbitrary random variables.

In a bounded open region of the $d$ dimensional space we consider a Brownian motion which is reborn at a fixed interior point as soon as it reaches the boundary. The evolution is invariant with respect to a density equal, modulo a constant, to the Green function of the Dirichlet Laplacian centered at the point of return. We calculate the resolvent in closed form, study its spectral properties and determine explicitly the spectrum in dimension one. Two proofs of the exponential ergodicity are given, one using the inverse Laplace transform and properties of analytic semigroups, and the other based on Doeblin's condition. Both methods admit generalizations to a wide class of processes.

We consider the nearest-neighbor simple random walk on $Z^d$, $d\ge2$, driven by a field of i.i.d. random nearest-neighbor conductances $\omega_{xy}\in[0,1]$. Apart from the requirement that the bonds with positive conductances percolate, we pose no restriction on the law of the $\omega$'s. We prove that, for a.e. realization of the environment, the path distribution of the walk converges weakly to that of non-degenerate, isotropic Brownian motion. The quenched functional CLT holds despite the fact that the local CLT may fail in $d\ge5$ due to anomalously slow decay of the probability that the walk returns to the starting point at a given time.

We consider a superprocess in a random environment represented by a random measure which is white in time and colored in space with correlation kernel $g(x,y)$. Suppose that $g(x,y)$ decays at a rate of $|x-y|^{-\alpha}$, $0\leq \alpha\leq 2$, as $|x-y|\to\infty$. We show that the process, starting from Lebesgue measure, suffers longterm local extinction. If $\alpha < 2$, then it even suffers finite time local extinction. This property is in contrast with the classical super-Brownian motion which has a non-trivial limit when the spatial dimension is higher than 2. We also show in this paper that in dimensions $d=1,2$ superprocess in random environment suffers local extinction for any bounded function $g$.

We consider the Poisson Boolean model of continuum percolation with balls of fixed radius $R$ in $n$-dimensional hyperbolic space $H^n$. Let $\lambda$ be the intensity of the underlying Poisson process, and let $N_C$ denote the number of unbounded components in the covered region. For the model in any dimension we show that there are intensities such that $N_C=\infty$ a.s. if $R$ is big enough. In $H^2$ we show a stronger result: for any $R$ there are two intensities $\lambda_c$ and $\lambda_u$ where $0< \lambda_c < \lambda _u < \infty$, such that$N_C=0$ for $\lambda \in [0,\lambda_c]$, $N_C=\infty$ for $\lambda \in (\lambda_c,\lambda_u)$ and $N_C=1$ for $\lambda \in [\lambda_u, \infty)$.

Let $\{X_1,\cdots ,X_N\}$ be a set of $N$ independent random variables, and let $S_n$ be a sum of $n$ random variables chosen without replacement from the set $\{X_1, \cdots , X_N\}$ with equal probabilities. In this paper we give a one-term Edgeworth expansion of the remainder term for the normal approximation of $S_n$ under mild conditions.

We consider Backward Stochastic Differential Equations (BSDEs) with generators that grow quadratically in the control variable. In a more abstract setting, we first allow both the terminal condition and the generator to depend on a vector parameter $x$. We give sufficient conditions for the solution pair of the BSDE to be differentiable in $x$. These results can be applied to systems of forward-backward SDE. If the terminal condition of the BSDE is given by a sufficiently smooth function of the terminal value of a forward SDE, then its solution pair is differentiable with respect to the initial vector of the forward equation. Finally we prove sufficient conditions for solutions of quadratic BSDEs to be differentiable in the variational sense (Malliavin differentiable).

We investigate unimodular random networks. Our motivations include their characterization via reversibility of an associated random walk and their similarities to unimodular quasi-transitive graphs. We extend various theorems concerning random walks, percolation, spanning forests, and amenability from the known context of unimodular quasi-transitive graphs to the more general context of unimodular random networks. We give properties of a trace associated to unimodular random networks with applications to stochastic comparison of continuous-time random walk.

We construct a stochastic process whose drift is a function of the process's local time at a reflecting barrier. The process arose as a model of the interactions of a Brownian particle and an inert particle in a paper by Knight [7]. We construct and give asymptotic results for two different arrangements of inert particles and Brownian particles, and construct the analogous process in higher dimensions.

We examine the total number of collisions $C_n$ in the $\Lambda$-coalescent process which starts with $n$ particles. A linear growth and a stable limit law for $C_n$ are shown under the assumption of a power-like behaviour of the measure $\Lambda$ near $0$ with exponent $0<\alpha<1$.

In this paper, a generalized It${\hat {\rm o}}$'s formula for continuous functions of two-dimensional continuous semimartingales is proved. The formula uses the local time of each coordinate process of the semimartingale, the left space first derivatives and the second derivative $\nabla _1^- \nabla _2^-f$, and the stochastic Lebesgue-Stieltjes integrals of two parameters. The second derivative $\nabla _1^- \nabla _2^-f$ is only assumed to be of locally bounded variation in certain variables. Integration by parts formulae are asserted for the integrals of local times. The two-parameter integral is defined as a natural generalization of both the Ito integral and the Lebesgue-Stieltjes integral through a type of It${\hat {\rm o }}$ isometry formula.

Brownian areas are considered in this paper: the Brownian excursion area, the Brownian bridge area, the Brownian motion area, the Brownian meander area, the Brownian double meander area, the positive part of Brownian bridge area, the positive part of Brownian motion area. We are interested in the asymptotics of the right tail of their density function. Inverting a double Laplace transform, we can derive, in a mechanical way, all terms of an asymptotic expansion. We illustrate our technique with the computation of the first four terms. We also obtain asymptotics for the right tail of the distribution function and for the moments. Our main tool is the two-dimensional saddle point method.