We investigate the probabilities of large deviations for the position of the front in a stochastic model of the reaction X+Y→2X on the integer lattice in which Y particles do not move while X particles move as independent simple continuous time random walks of total jump rate 2. For a wide class of initial conditions, we prove that a large deviations principle holds and we show that the zero set of the rate function is the interval [0, v], where v is the velocity of the front given by the law of large numbers. We also give more precise estimates for the rate of decay of the slowdown probabilities. Our results indicate a gapless property of the generator of the process as seen from the front, as it happens in the context of nonlinear diffusion equations describing the propagation of a pulled front into an unstable state.

We consider a stochastic billiard in a random tube which stretches to infinity in the direction of the first coordinate. This random tube is stationary and ergodic, and also it is supposed to be in some sense well behaved. The stochastic billiard can be described as follows: when strictly inside the tube, the particle moves straight with constant speed. Upon hitting the boundary, it is reflected randomly, according to the cosine law: the density of the outgoing direction is proportional to the cosine of the angle between this direction and the normal vector. We also consider the discrete-time random walk formed by the particle’s positions at the moments of hitting the boundary. Under the condition of existence of the second moment of the projected jump length with respect to the stationary measure for the environment seen from the particle, we prove the quenched invariance principles for the projected trajectories of the random walk and the stochastic billiard.

For a class of stochastic differential equations with reflection for which a certain ${\mathbb{L}}^{p}$ continuity condition holds with p>1, it is shown that any weak solution that is a strong Markov process can be decomposed into the sum of a local martingale and a continuous, adapted process of zero p-variation. When p=2, this implies that the reflected diffusion is a Dirichlet process. Two examples are provided to motivate such a characterization. The first example is a class of multidimensional reflected diffusions in polyhedral conical domains that arise as approximations of certain stochastic networks, and the second example is a family of two-dimensional reflected diffusions in curved domains. In both cases, the reflected diffusions are shown to be Dirichlet processes, but not semimartingales.

A complete representation of the Martin boundary of killed random walks on the quadrant ℕ^*×ℕ^* is obtained. It is proved that the corresponding full Martin compactification of the quadrant ℕ^*×ℕ^* is homeomorphic to the closure of the set {w=z/(1+|z|) : z∈ℕ^*×ℕ^*} in ℝ². The method is based on a ratio limit theorem for local processes and large deviation techniques.

We prove an existence and uniqueness result for the obstacle problem of quasilinear parabolic stochastic PDEs. The method is based on the probabilistic interpretation of the solution by using the backward doubly stochastic differential equation.

For 0<α≤2, a super-α-stable motion X in $\mathsf{R}^{d}$ with branching of index 1+β∈(1, 2) is considered. Fix arbitrary t>0. If d<α/β, a dichotomy for the density function of the measure X_t holds: the density function is locally Hölder continuous if d=1 and α>1+β but locally unbounded otherwise. Moreover, in the case of continuity, we determine the optimal local Hölder index.

We use orthogonality measures of Askey–Wilson polynomials to construct Markov processes with linear regressions and quadratic conditional variances. Askey–Wilson polynomials are orthogonal martingale polynomials for these processes.

We show that in the point process limit of the bulk eigenvalues of β-ensembles of random matrices, the probability of having no eigenvalue in a fixed interval of size λ is given by

(κ_β+o(1))λ^γ_β exp((−β/64)λ²+(β/8−1/4)λ)

as λ→∞, where

γ_β=1/4(β/2+2/β−3)

and κ_β is an undetermined positive constant. This is a slightly corrected version of a prediction by Dyson [J. Math. Phys. 3 (1962) 157–165]. Our proof uses the new Brownian carousel representation of the limit process, as well as the Cameron–Martin–Girsanov transformation in stochastic calculus.

We study the Poissonnian ensembles of Markov loops and the associated renormalized self intersection local times.

A generalized version of a well-known problem of D. G. Kendall states that the zero cell of a stationary Poisson hyperplane tessellation in ℝ^d, under the condition that it has large volume, approximates with high probability a certain definite shape, which is determined by the directional distribution of the underlying hyperplane process. This result is extended here to typical k-faces of the tessellation, for k∈{2, …, d−1}. This requires the additional condition that the direction of the face be in a sufficiently small neighbourhood of a given direction.