The product Bernoulli measures $\nu_\alpha$ with densities $\alpha$, $\alpha\in [0,1]$, are the extremal translation invariant stationary measures for an exclusion process on $\mathbb{Z}$ with irreducible random walk kernel $p(\cdot)$. Stationary measures that are not translation invariant are known to exist for finite range $p(\cdot)$ with positive mean. These measures have particle densities that tend to 1 as $x\to\infty$ and tend to 0 as $x\to -\infty$; the corresponding extremal measures form a one-parameter family and are translates of one another. Here, we show that for an exclusion process where $p(\cdot)$ is irreducible and has positive mean, there are no other extremal stationary measures. When $\sum_{x<0} x^2 p(x) =\infty$, we show that any nontranslation invariant stationary measure is not a blocking measure; that is, there are always either an infinite number of particles to the left of any site or an infinite number of empty sites to the right of the site. This contrasts with the case where $p(\cdot)$ has finite range and the above stationary measures are all blocking measures. We also present two results on the existence of blocking measures when $p(\cdot)$ has positive mean, and $p(y)\leq p(x)$ and $p(-y)\leq p(-x)$ for $1\leq x\leq y$. When the left tail of $p(\cdot)$ has slightly more than a third moment, stationary blocking measures exist. When $p(-x)\leq p(x)$ for $x>0$ and $\sum_{x<0}x^2p(x)>\infty$, stationary blocking measures also exist.

We consider the supremum $\mathcal{W}_n$ of self-normalized empirical processes indexed by unbounded classes of functions $\mathcal{F}$. Such variables are of interest in various statistical applications, for example, the likelihood ratio tests of contamination. Using the Herbst method, we prove an exponential concentration inequality for $\mathcal{W}_n$ under a second moment assumption on the envelope function of $\mathcal{F}$. This inequality is applied to obtain moderate deviations for $\mathcal{W}_n$. We also provide large deviations results for some unbounded parametric classes $\mathcal{F}$.

We equip the intersection of p independent Brownian paths in $\mathbb{R}^d$, $d\ge 2$, with the natural measure $\ell$ defined by projecting the intersection local time measure via one of the Brownian motions onto the set of intersection points. Given a bounded domain $U\subset\mathbb{R}^d$ we show that, as $a\uparrow\infty$, the probability of the event $\{\ell(U)>a\}$ decays with an exponential rate of $a^{1/p}\theta$, where $\theta$ is described in terms of a variational problem. In the important special case when U is the unit ball in $\mathbb{R}^3$ and $p=2$, we characterize $\theta$ in terms of an ordinary differential equation. We apply our results to the problem of finding the Hausdorff dimension spectrum for the thick points of the intersection of two independent Brownian paths in $\mathbb{R}^3$.

Under minimal moment conditions complete asymptotic expansions are obtained for expectations of unbounded functions of a finite family of independent random variables in the Poissonian setting when the distributions of the random variables have large atoms at zero.

We study a pair of populations in $\mathbb{R}^{2}$ which undergo diffusion and branching. The system is interactive in that the branching rate of each type is proportional to the local density of the other type. For a diffusion rate sufficiently large compared with the branching rate, the model is constructed as the unique pair of finite measure-valued processes which satisfy a martingale problem involving the collision local time of the solutions. The processes are shown to have densities at fixed times which live on disjoint sets and explode as they approach the interface of the two populations. In the long-term limit, global extinction of one type is shown. The process constructed is a rescaled limit of the corresponding $\mathbb{Z}^{2}$-lattice model studied by D. A. Dawson and E. A. Perkins and resolves the large scale mass-time-space behavior of that model.

The aim of the present paper is to construct a stochastic process, whose law is the solution of the Smoluchowski's coagulation equation. We introduce first a modified equation, dealing with the evolution of the distribution $Q_t(dx)$ of the mass in the system. The advantage we take on this is that we can perform an unified study for both continuous and discrete models.

The integro-partial-differential equation satisfied by $\{Q_t\}_{t\geq 0}$ can be interpreted as the evolution equation of the time marginals of a Markov pure jump process. At this end we introduce a nonlinear Poisson driven stochastic differential equation related to the Smoluchowski equation in the following way: if $X_t$ satisfies this stochastic equation, then the law of $X_t$ satisfies the modified Smoluchowski equation. The nonlinear process is richer than the Smoluchowski equation, since it provides historical information on the particles.

Existence, uniqueness and pathwise behavior for the solution of this SDE are studied. Finally, we prove that the nonlinear process X can be obtained as the limit of a Marcus-Lushnikov procedure.

In this paper, we give a probabilistic interpretation of a viscous scalar conservation law in a bounded interval thanks to a nonlinear martingale problem. The underlying nonlinear stochastic process is reflected at the boundary to take into account the Dirichlet conditions. After proving uniqueness for the martingale problem, we show existence thanks to a propagation of chaos result. Indeed we exhibit a system of N interacting particles, the empirical measure of which converges to the unique solution of the martingale problem as $N\to+\infty$. As a consequence, the solution of the viscous conservation law can be approximated thanks to a numerical algorithm based on the simulation of the particle system. When this system is discretized in time thanks to the Euler-Lépingle scheme, we show that the rate of convergence of the error is in $\OO(\Delta t +1/\sqrt{N})$, where $\Delta t$ denotes the time step. Finally, we give numerical results which confirm this theoretical rate.

Let $f_{i}$, $i=1,\dots,n$, be copies of a random variable f and let N be an Orlicz function. We show that for every $x\in \mathbb{R}^{n}$ the expectation $\mathbf{E} \| (x_i f_i) _{i=1}^n \|_N $ is maximal (up to an absolute constant) if $f _{i}$, $i=1,\dots,n$, are independent. In that case we show that the expectation $\mathbf{E} \| (x_i f_i) _{i=1}^n \| _N $ is equivalent to $\|x\| _M$, for some Orlicz function M depending on N and on distribution of f only. We provide applications of this result.

The maxima of partial sums indexed by squares and rectangles over lattice points and random cubes are studied in this paper. For some of these problems, the dimension ($d=1, d=2$ and $d \geq 3$) significantly affects the limit behavior of the maxima. However, for other problems, the maxima behave almost the same as their one-dimensional counterparts. The tools for proving these results are large deviations, the Chen-Stein method, number theory and inequalities of empirical processes.

Let $\{X_i;\, i\geq 1\},\,\{Y_i;\,i\geq 1\},\,\{U, U_i;\, i\geq 1\}$ and $\{V, V_i;\, i\geq 1\}$ be four i.i.d. sequences of random variables. Suppose U and V are uniformly distributed on $[0,1]^3.$ For each realization of $\{U_j;\, 1\leq j\leq n\},\ \{X_{i,p};\break \, 1\leq p \leq n\}$ is constructed as a certain permutation of $\{X_p;\, 1\leq p\leq n\}$ for any $1\leq i \leq n.$ Also, $\{Y_{j,p};\, 1\leq p \leq n\}, 1\leq j\leq n,$ are constructed the same way, based on $\{Y_j\}$ and $\{V_j\}.$ For a score function F, we show that \begin{eqnarray*} W_n:= \max_{1\leq i, j,m \leq n}\sum_{p=1}^m F(X_{i,p},Y_{j, p}) \end{eqnarray*} has an asymptotic extreme distribution with the same parameters as in the one-dimensional case. This model is constructed for a comparison of scores of protein structures with foldings.

We consider the symmetric simple exclusion process on $\ZZ^d$, for $d\geq 5$, and study the regularity of the quasi-stationary measures of the dynamics conditioned on not occupying the origin. For each $\rho\in ]0,1[$, we establish uniqueness of the density of quasi-stationary measures in $L^2(d\nur)$, where $\nur$ is the stationary measure of density $\rho$. This, in turn, permits us to obtain sharp estimates for $P_{\nur}(\tau>t)$, where $\tau$ is the first time the origin is occupied.

A general set of distribution-free conditions is described under which an i.i.d. sequence of random variables is preserved under optional skipping. This work is motivated by theorems of J. L. Doob and Z. Ignatov, unifying and extending aspects of both.

We consider nonconservative, reversible spin systems, with unbounded discrete spins. We show that for a class of these dynamics in a high temperature regime, the relative entropy with respect to the equilibrium distribution decays exponentially in time, although the logarithmic-Sobolev inequality fails. To this end we prove a weaker modification of the logarithmic-Sobolev inequality.

We consider a Brownian bridge on the hyperbolic plane with one extremity tending to infinity, in finite time. We show that the exact exponential rate according to which this process concentrates round the geodesical segment joining the origin o to the moving extremity z is $-\rho(o,z)$, where $\rho$ stands for the hyperbolic distance. This improves a result of A. Eberle.

A typical feature of the long time behavior of continuous super-Brownian motion in a stable catalytic medium is the development of large mass clumps (or clusters) at spatially rare sites. We describe this phenomenon by means of a functional limit theorem under renormalization. The limiting process is a Poisson point field of mass clumps with no spatial motion component and with infinite variance. The mass of each cluster evolves independently according to a non-Markovian continuous process trapped at mass zero, which we describe explicitly by means of a Brownian snake construction in a random medium. We also determine the survival probability and asymptotic size of the clumps.

We consider the way sets are dispersed by the action of stochastic flows derived from martingale fields. Under fairly general continuity and ellipticity conditions, the following dichotomy result is shown: any nontrivial connected set $\mathcal{X}$ either contracts to a point under the action of the flow, or its diameter grows linearly in time, with speed at least a positive deterministic constant $\gL$. The linear growth may further be identified (again, almost surely), with a much stronger behavior, which we call "ball-chasing'': if $\psi$ is any path with Lipschitz constant smaller than $\gL$, the ball of radius $\gep$ around $\psi(t)$ contains points of the image of $\mathcal{X}$ for an asymptotically positive fraction of times $t$. If the ball grows as the logarithm of time, there are individual points in $\mathcal{X}$ whose images eventually remain in the ball.

We consider the standard Abelian sandpile process on the Bethe lattice. We show the existence of the thermodynamic limit for the finite volume stationary measures and the existence of a unique infinite volume Markov process exhibiting features of self-organized criticality.