We use the exploration process introduced in a previous work to develop a new construction of superprocesses with a general branching mechanism. This construction depends on a path-valued process called the Lévy snake, which is of independent interest. Our method of proof involves a calculation of the Laplace functional of the occupation field of the Lévy snake. This calculation relies on an evaluation of the corresponding moment functionals, which requires precise information about the underlying genealogical structure.

We study two typical examples of countable Markov chains in random environment using the Lyapunov functions method: random walk and random string in random environment. In each case we construct an explicit Lyapunov function. Investigating the behavior of this function, we get the classification for recurrence, transience, ergodicity. We obtain new results for random strings in random environment, though we simply review well-known results for random walks using our approach.

Assign to the lattice sizes $z \epsilon \mathbb{Z}^d$ i.i.d. random 2 $d$-dimensional vectors $(\omega(z, z + e))_{|e|=1}$ whose entries take values in the open unit interval and add up to one. Given a realization $\omega$ of this environment, let $(X_n)_{n \geq o}$ be a Markov chain on $\mathbb{Z}^d$ which, when at $z$, moves one step to its neighbor $z + e$ with transition probability $\omega(z, z + e)$. We derive a large deviation principle for $X_n/n$ by means of a result similar to the shape theorem of first-passage percolation and related models. This result produces certain constants that are the analogue of the Lyapounov exponents known from Brownian motion in Poissonian potential or random walk in random potential. We follow a strategy similar to Sznitman.

We study the sample path asymptotics of a class of recurrent diffusion processes with random potentials, including examples of Sinai’s simple random walk in random environment and Brox’s diffusion process with Brownian potential. The main results consist of several integral criteria which completely characterize all the possible Lévy classes, therefore providing a very precise image of the almost sure asymptotic behaviors of these processes.

We consider backward stochastic differential equations with convex constraints on the gains (or intensity-of-noise) process. Existence and uniqueness of a minimal solution are established in the case of a drift coefficient which is Lipschitz continuous in the state and gains processes and convex in the gains process. It is also shown that the minimal solution can be characterized as the unique solution of a functional stochastic control-type equation. This representation is related to the penalization method for constructing solutions of stochastic differential equations, involves change of measure techniques, and employs notions and results from convex analysis, such as the support function of the convex set of constraints and its various properties.

The optimal nonlinear filtering problem for a diffusion process in a noncompact domain, observed in white noise, is considered. It is assumed that the process is ergodic, the diffusion coefficient is constant and the observation is linear. Using known bounds on the conditional density, it is shown that when the observation noise is sufficiently small, the filter is exponentially stable, and that the decay rate of the total variation distance between differently initialized filtering processes tends to infinity as the noise intensity approaches zero.

We first show a large deviation principle for degenerate diffusion-transmutation processes and study the Riemannian metric associated with the action functional under a Hörmander-type assumption. Then we study the behavior of the solution $u^{\varepsilon}$ of a system of strongly coupled scaled KPP equations. Using backward stochastic differential equations and the theory of Hamilton-Jacobi equations, we show that, when the parabolic operator satisfies a Hörmander-type hypothesis or when the nonlinearity depends on the gradient, the wavefront location is given by the same formula as that in Freidlin and Lee or Barles, Evans and Souganidis. We obtain the exact logarithmic rates of convergence to the unstable equilibrium state in the general case and to the stable equilibrium state when the equations are uniformly positively coupled.

The solution is found to the optimal stopping problem with payoff $$\sup_{\tau} E(S_{\tau} - \int_0^{\tau} c(X_t)dt),$$ where $S = (S_t)_{t \geq 0}$ is the maximum process associated with the one-dimensional time-homogeneous diffusion $X = (X_t)_{t \geq 0}$, the function $x \mapsto c(x)$ is positive and continuous, and the supremum is taken over all stopping times $\tau$ of $X$ for which the integral has finite expectation. It is proved, under no extra conditions, that this problem has a solution; that is, the payoff is finite and there is an optimal stopping time, if and only if the following maximality principle holds: the first-order nonlinear differential equation $$g'(s) = \frac{\sigma^2 (g(s))L'(g(s))}{2c(g(s))(L(s) - L(g(s)))}$$ admits a maximal solution $s \mapsto g_*(s)$ which stays strictly below the diagonal in $\mathbb{R}^2$. [In this equation $x \mapsto \sigma(x)$ is the diffusion coefficient and $x \mapsto L(x)$ the scale function of $X$.] In this case the stopping time $$\tau_* = \inf{t > 0|X_t \leq g_*(S_t)}$$ is proved optimal, and explicit formulas for the payoff are given. The result has a large number of applications and may be viewed as the cornerstone in a general treatment of the maximum process.

In this paper we show that the variational representation $$-\log Ee^{-f(W)} = \inf_v E{1/2 \int_0^1 \parallel v_s \parallel^2 ds + f(W + \int_0^{\cdot} v_s ds)}$$ holds, where $W$ is a standard $d$-dimensional Brownian motion, $f$ is any bounded measurable function that maps $C([0, 1]: \mathbb{R}^d)$ into $\mathbb{R}$ and the infimum is over all processes $v$ that are progressively measurable with respect to the augmentation of the filtration generated by $W$. An application is made to a problem concerned with large deviations, and an extension to unbounded functions is given.

A universal large deviation lower bound is proved for sums of Banach space valued functions of an irreducible, general state space Markov chain. There are no restrictions on the functions (other than measurability).

An identity in distribution due to Knight for Brownian motion is extended in two different ways: first by replacing the supremum of a reflecting Brownian motion by the range of an unreflected Brownian motion and second by replacing the reflecting Brownian motion by a recurrent Bessel process. Both extensions are explained in terms of random Brownian scaling transformations and Brownian excursions. The first extension is related to two different constructions of Itô’s law of Brownian excursions, due to Williams and Bismut, each involving back-to-back splicing of fragments of two independent three-dimensional Bessel processes. Generalizations of both splicing constructions are described, which involve Bessel processes and Bessel bridges of arbitrary positive real dimension.

Regard an element of the set $$\Delta := {(x_1, x_2, \dots): x_1 \geq x_2 \geq \dots \geq 0, \Sigma_i x_i = 1}$$ as a fragmentation of unit mass into clusters of masses $x_i$. The additive coalescent of Evans and Pitman is the $\Delta$-valued Markov process in which pairs of clusters of masses ${x_i, x_j}$ merge into a cluster of mass $x_i + x_j$ at rate $x_i + x_j$. They showed that a version $(\rm X^{\infty}(t), -\infty < t < \infty)$ of this process arises as a $n \to \infty$ weak limit of the process started at time $-1/2 \log n$ with $n$ clusters of mass $1/n$. We show this standard additive coalescent may be constructed from the continuum random tree of Aldous by Poisson splitting along the skeleton of the tree. We describe the distribution of $\rm X^{\infty}(t)$ on $\Delta$ at a fixed time $t$. We show that the size of the cluster containing a given atom, as a process in $t$, has a simple representation in terms of the stable subordinator of index 1/2. As $t \to -\infty$, we establish a Gaussian limit for (centered and normalized) cluster sizes and study the size of the largest cluster.

In connection with a conjecture stated by D. G. Kendall in the forties, we describe the asymptotic behavior of the distribution function of the area of the planar Crofton cell. We deduce from this (in support of his conjecture) that expressed in terms of eigenvalues, the large Crofton cells are nearly circular. We obtain also the asymptotic behavior of the Laplace transform of the law of the perimeter of the convex hull of planar Brownian motion run until time 1. This last result implies that the small convex hulls of Brownian motion are nearly circular.

Approximating a critical attractive reversible nearest particle system on a finite set from above is not difficult, but approximating it from below is less trivial, as the empty configuration is invariant. We develop a finite state Markov chain that deals with this issue. The rate of convergence for this chain is discovered through a mixing inequality in Jerrum and Sinclair; an application of that spectral gap bound in this case requires the use of ‘‘randomized paths from state to state.’’ For applications, we prove distributional results for semiinfinite and infinite critical RNPS.

Benjamini, Pemantle and Peres constructed nearest neighbor processes which have predictability profiles that decay faster than that of the simple random walk. Häggström and Mossel found processes with even faster decaying predictability profiles. We prove that the rate of decay achieved by Häggström and Mossel is optimal.

.Any infinite graph $G = (V, E)$ has a site percolation critical probability $p_c^{\rm site}$ and a bond percolation critical probability $p_c^{\rm bond}$. The well-known weak inequality $p_c^{\rm site} \geq p_c^{\rm bond}$ is strengthened to strict inequality for a c c broad category of graphs $G$, including all the usual finite-dimensional lattices in two and more dimensions. The complementary inequality $p_c^{\rm site} \leq 1 - (1 - p_c^{\rm bond})^{\Delta - 1}$ is proved also, where $\Delta$ denotes the supremum of the vertex degrees of $G$.

Let $\mathscr{C}$ be the set of chambers of a real hyperplane arrangement. We study a random walk on $\mathscr{C}$ introduced by Bidigare, Hanlon and Rockmore. This includes various shuffling schemes used in computer science, biology and card games. It also includes random walks on zonotopes and zonotopal tilings. We find the stationary distributions of these Markov chains, give good bounds on the rate of convergence to stationarity, and prove that the transition matrices are diagonalizable. The results are extended to oriented matroids.

We determine the logarithmic Sobolev constant for the Bernoulli- Laplace model and the time to stationarity for the symmetric simple exclusion model up to the leading order. Our method for proving the logarithmic Sobolev inequality is based on a martingale approach and is applied to the random transposition model as well. The proof for the time to stationarity is based on a general observation relating the time to stationarity to the hydrodynamical limit.

The hydrodynamic limit of the symmetric simple exclusion process with speed change is given by a diffusive equation in the appropriate scale. Following the nongradient method introduced by Varadhan and the Navier-Stokes methods developed by Yau, we prove that in the same scale, the next order correction is given by a third order equation for dimension $d \geq 3$.