Solutions of semilinear elliptic differential equations in infinite-dimensional spaces are obtained by means of forward and backward infinite-dimensional stochastic evolution equations. The backward equation is considered on an infinite time horizon and a suitable growth condition replaces the final condition. Elliptic equations are intended in a mild sense, suitable also for applications to optimal control. We finally notice that, due to the lack of smoothing properties, the elliptic partial differential equation considered here could not be treated by analytic methods.

We prove a large deviations principle for the occupation time of a site in the two-dimensional symmetric simple exclusion process. The decay probability rate is of order $t/\log t$ and the rate function is given by $\Upsilon_\alpha (\beta) = (\pi/2) \{\sin^{-1}(2\beta-1)-\sin^{-1}(2\alpha -1) \}^2$. The proof relies on a large deviations principle for the polar empirical measure which contains an interesting $\log$ scale spatial average. A contraction principle permits us to deduce the occupation time large deviations from the large deviations for the polar empirical measure.

We consider the $(d+1)$-dimensional an dynamical system constituted by weakly coupled expanding circle maps on $\Z^d$ together with the spatial shifts. This viewpoint allows us to use thermodynamic formalism, and to describe the asymptotic behavior of the system in this setup. We obtain a volume lemma, which describes the exponential behavior of the size under Lebesgue measure of dynamical balls around any orbit, and then a large deviations principle for the empirical measure associated to this dynamical system. The proofs are direct: we do not use the coding constructed by Jiang for such systems.

Scaling limits of continuous time random walks are used in physics to model anomalous diffusion, in which a cloud of particles spreads at a different rate than the classical Brownian motion. Governing equations for these limit processes generalize the classical diffusion equation. In this article, we characterize scaling limits in the case where the particle jump sizes and the waiting time between jumps are dependent. This leads to an efficient method of computing the limit, and a surprising connection to fractional derivatives.

It is shown that the product of the Rényi entropies of two independent random vectors provides a sharp lower bound for the expected value of the moments of the inner product of the random vectors. This new inequality contains important geometry (such as extensions of one of the fundamental affine isoperimetric inequalities, the Blaschke--Santaló inequality).

The paper is concerned with reflecting Brownian motion (RBM) in domains with deterministic moving boundaries, also known as "noncylindrical domains,'' and its connections with partial differential equations. Construction is given for RBM in $C^3$-smooth time-dependent domains in the n-dimensional Euclidean space $\R^n$. We present various sample path properties of the process, two-sided estimates for its transition density function, and a probabilistic representation of solutions to some partial differential equations. Furthermore, the one-dimensional case is thoroughly studied, with the assumptions on the smoothness of the boundary drastically relaxed.

We consider an arbitrary one-dimensional conservative particle system with finite-range interactions and finite site capacity, governed on the hydrodynamic scale by a scalar conservation law with Lipschitz-continuous flux h. A finite-size perturbation restricts the local current to some maximum value $\phi$. We show that the perturbed hydrodynamic behavior is entirely determined by $\phi$ if $\inf(h;\phi)$ is first nondecreasing and then nonincreasing, which we believe is a necessary condition.

We consider the simple asymmetric exclusion process with nonzero drift under the stationary Bernoulli product measure at density $\rho$. We prove that for dimension $d=2$ the occupation time of the site 0 is diffusive as soon as $\rho\neq 1/2$. For dimension $d=1$, if the density $\rho$ is equal to $1/2$, we prove that the time t variance of the occupation time of the site 0 diverges in a certain sense at least as $t^{5/4}$.

We prove a law of large numbers for a class of ballistic, multidimensional random walks in random environments where the environment satisfies appropriate mixing conditions, which hold when the environment is a weak mixing field in the sense of Dobrushin and Shlosman. Our result holds if the mixing rate balances moments of some random times depending on the path. It applies in the nonnestling case, but we also provide examples of nestling walks that satisfy our assumptions. The derivation is based on an adaptation, using coupling, of the regeneration argument of Sznitman and Zerner.

We consider a Markov chain on the space of (countable) partitions of the interval $[0,1]$, obtained first by size-biased sampling twice (allowing repetitions) and then merging the parts (if the sampled parts are distinct) or splitting the part uniformly (if the same part was sampled twice). We prove a conjecture of Vershik stating that the Poisson--Dirichlet law with parameter $\theta=1$ is the unique invariant distribution for this Markov chain. Our proof uses a combination of probabilistic, combinatoric and representation-theoretic arguments.

This paper proves that the scaling limit of a loop-erased random walk in a simply connected domain $Dsubsetneqq\C$ is equal to the radial SLE$_2$ path. In particular, the limit exists and is conformally invariant. It follows that the scaling limit of the uniform spanning tree in a Jordan domain exists and is conformally invariant. Assuming that $\p D$ is a $C^1$-simple closed curve, the same method is applied to show that the scaling limit of the uniform spanning tree Peano curve, where the tree is wired along a proper arc $A\subset\p D$, is the chordal SLE$_8$ path in $\overline D$ joining the endpoints of A. A by-product of this result is that SLE$_8$ is almost surely generated by a continuous path. The results and proofs are not restricted to a particular choice of lattice.

Suppose that the integers are assigned the random variables $\{\omega_x,\mu_x\}$ (taking values in the unit interval times the space of probability measures on $\reals_+$), which serve as an environment. This environment defines a random walk $\{X_t\}$ (called a RWREH) which, when at x, waits a random time distributed according to $\mu_x$ and then, after one unit of time, moves one step to the right with probability $\omega_x$, and one step to the left with probability $1-\omega_x$. We prove large deviation principles for $X_t/t$, both quenched (i.e., conditional upon the environment), with deterministic rate function, and annealed (i.e., averaged over the environment). As an application, we show that for random walks on Galton--Watson trees, quenched and annealed rate functions along a ray differ.

We study a new type of representation problem for optional processes with connections to singular control, optimal stopping and dynamic allocation problems. As an application, we show how to solve a variant of Skorohod's obstacle problem in the context of backward stochastic differential equations.

We study a class of linear hyperbolic stochastic partial differential equations in bounded domains, which includes the wave equation and the telegraph equation, driven by Gaussian noise that is white in time but not in space. We give necessary and sufficient conditions on the spatial correlation of the noise for the existence (and uniqueness) of square-integrable solutions. In the particular case where the domain is a ball and the noise is concentrated on a sphere, we characterize the isotropic Gaussian noises with this property. We also give explicit necessary and sufficient conditions when the domain is a hypercube and the Gaussian noise is concentrated on a hyperplane.

Following classical work by Freidlin [Trans. Amer. Math. Soc. (1988) 305 665--657] and subsequent works by Sowers [Ann. Probab. (1992) 20 504--537] and Peszat [Probab. Theory Related Fields (1994) 98 113--136], we prove large deviation estimates for the small noise limit of systems of stochastic reaction--diffusion equations with globally Lipschitz but unbounded diffusion coefficients, however, assuming the reaction terms to be only locally Lipschitz with polynomial growth. This generalizes results of the above mentioned authors. Our results apply, in particular, to systems of stochastic Ginzburg--Landau equations with multiplicative noise.

In the early 1940s, D. G. Kendall conjectured that the shape of the zero cell of the random tessellation generated by a stationary and isotropic Poisson line process in the plane tends to circularity given that the area of the zero cell tends to $\infty$. A proof was given by I. N. Kovalenko in 1997. This paper generalizes Kovalenko's result in two directions: to higher dimensions and to not necessarily isotropic stationary Poisson hyperplane processes. In the anisotropic case, the asymptotic shape of the zero cell depends on the direction distribution of the hyperplane process and is obtained from it via an application of Minkowski's existence theorem for convex bodies with given area measures.

This paper is concerned with a stochastic model for the spread of an epidemic among a population of n individuals, labeled $1,2,\ldots,n$, in which a typical infected individual, i say, makes global contacts, with individuals chosen independently and uniformly from the whole population, and local contacts, with individuals chosen independently according to the contact distribution ${V_{i}^{n} = \{ v_{i,j}^{n} ; j=1,2, \ldots, n \}}$, at the points of independent Poisson processes with rates $\lambda_G^{n}$ and $\lambda_L^{n}$, respectively, throughout an infectious period that follows an arbitrary but specified distribution. The population initially comprises $m_n$ infectives and $n-m_n$ susceptibles. A sufficient condition is derived for the number of individuals who survive the epidemic to converge weakly to a Poisson distribution as $n \to \infty$. The result is specialized to the households model, in which the population is partitioned into households and local contacts are chosen uniformly within an infective's household; the overlapping groups model, in which the population is partitioned in several ways and local mixing is uniform within the elements of the partitions; and the great circle model, in which $v_{i,j}^{n} = v_{(i-j)_{\mod n}}^{n}$.