Degenerate white noise perturbations of Hamiltonian systems in $R^2$ are studied. In particular, perturbations of a nonlinear oscillator with 1 degree of freedom are considered. If the oscillator has more than one stable equilibrium, the long time behavior of the system is defined by a diffusion process on a graph. Inside the edges the process is defined by a standard averaging procedure, but to define the process for all $t > 0$ one should add gluing conditions at the vertices. Calculation of the gluing conditions is based on delicate Hörmander-type estimates.

The uniqueness in law for the equation $\partial X_t/ \partial t = 1/2 \delta X_t + X_t^{\gamma} \dot{W}$ is established for $1/2 < \gamma < 1$. The proof uses a duality technique and requires the construction of an approximating sequence of dual processes.

Given a Riemannian manifold $M$ and loop $\phi: X^1 \mapsto M$, we construct a Gaussian random process $S^1 \ni \theta \leadsto X_{\theta} \epsilon T_{\phi(\theta)}M$, which is an analog of the Brownian motion process in the sense that the formal covariant derivative $\theta \leadsto \nabla_{\theta}X_{\theta}$ appears as a stationary process whose spectral measure is uniformly distributed over some discrete set. We show that $X$ satisfies the two-point Markov property reciprocal process if the holonomy along the loop $\phi$ is nontrivial. The covariance function of $X$ is calculated and the associated abstract Wiener space is described. We also characterize $X$ as a solution of a special nondiffusion type stochastic differential equation. Somewhat surprisingly, the nature of $X$ turns out to be very different if the holonomy along $\phi$ is the identity map $I: T_{\phi(0)}M \mapsto T_{\phi(0)}M$. In this case, we show that the usual periodic Ornstein-Uhlenbeck process, associated with a harmonic oscillator at nonzero temperature, may be viewed as a standard velocity process in which the driving Brownian motion is replaced by the process $X$.

We consider $d$-dimensional Brownian motion in a truncated Poissonian potential conditioned to reach a remote location. If Brownian motion starts at the origin and ends in an hyperplane at distance $L$ from the origin, the transverse fluctuation of the path is expected to be of order $L^{\xi}$ We are interested in a lower bound for $\xi$. We first show that $\xi \geq 1/2$ in dimensions $d \geq 2$ and then we prove superdiffusive behavior for $d = 2$, resulting in $\xi \geq 3/5$.

The simple random walk on a supercritical Galton–Watson tree is transient when the tree is infinite. Moreover, there exist regeneration times, that is, times when the walk crosses an edge for the first and the last time. We prove that the distance and the range of the walk satisfy functional central limit theorems under the annealed law GP. This result is a consequence of estimates of the law of the first regeneration time $\tau_R$. We show that there exist positive $c , c', \alpha$ and $\beta$ such that, for all $n \geq 1$, $$\exp (-cn^{\alpha}) \leq GP[\tau_R \geq \exp (-c' n^{\beta}).$

La marche au hasard simple sur un arbre de Galton–Watson super-critique est transiente quand l’arbre est infini. De plus, elle possède des instants de régénération, c’est- à-dire, des instants où le marcheur traverse une arête pour la première et la dernière fois. Nous démontrons que la distance et le rang de la marche vérifient un théorème central limite fonctionnel sous la loi GP moyennée en l’environnement. Ce résultat repose sur une estimation de la loi du premier instant de régénération $\tau_R$. Nous montrons qu’il existe des constantes $c , c', \alpha$ et $\beta$ strictement positives telles que, pour tout $n \geq 1$, $$\exp (-cn^{\alpha}) \leq GP[\tau_R \geq \exp (-c' n^{\beta}).$

We prove a Kolmogorov test for super-Brownian motion started at the Dirac mass at the origin. More precisely, we determine the functions $g$ such that for all $t$ small enough, the support of the process at time $t$ will be contained in the ball of radius $g(t)$ centered at 0. As a consequence, we get a necessary and sufficient condition for the existence in certain space-time domains of a solution of the associated semilinear partial differential equation that blows up at the origin.

We consider two spatial branching models on $\mathbb{R}^d$: branching Brownian motion with a branching law in the domain of normal attraction of a $1 + \beta$ stable law, $0 < \beta \leq 1$, and the corresponding high density limit measure valued diffusion. The longtime behavior of both models depends highly on $\beta$ and $d$. We show that for $d \leq \frac{2}{\beta}$ the only invariant measure is $\delta_0$, the unit mass on the empty configuration. Furthermore, we give a precise condition for convergence toward $\delta_0$. For $d > \frac{2}{\beta}$ it is known that there exists a family $(\nu_{\theta}, \theta \epsilon [0, \infty))$ of nontrivial invariant measures. We show that every invariant measure is a convex combination of the $\nu_{\theta}$. Both results have been known before only under an additional finite mean assumption. For the critical dimension $d = \frac{2}{\beta}$ we show that both models display the phenomenon of diffusive clustering. This means that clusters grow spatially on a random scale. We give a precise description of the clusters via multiple scale analysis. Our methods rely mainly on studying sub- and supersolutions of the reaction diffusion equation $\frac{\partial u}{\partial t} - 1/2 \Delta u + u^{1 + \beta} = 0$.

We study a system of two interacting populations which undergo random migration and mutually catalytic branching. The branching rate of one population at a site is proportional to the mass of the other population at the site. The system is modelled by an infinite system of stochastic differential equations, allowing symmetric Markov migration, if the set of sites is discrete $(\mathbb{Z}^d)$, or by a stochastic partial differential equation with Brownian migration if the set of sites is the real line. A duality technique of Leonid Mytnik, which gives uniqueness in law, is used to examine the long-time behavior of the solutions. For example, with uniform initial conditions, the process converges to an equilibrium distribution as $t \to \infty$, and there is coexistence of types in the equilibrium “iff ” the random migration is transient.

We study translation-invariant random-cluster measures with techniques from large deviation theory and convex analysis. In particular, we prove a large deviation principle with rate function given by a specific entropy, and a Dobrushin-Lanford-Ruelle variational principle that characterizes translation-invariant random-cluster measures as the solutions of the variational equation for free energy. Consequences of these theorems include inequalities for edge and cluster densities of translation-invariant random-cluster measures.

We extend a result of Freidlin and Gartner for KPP wave fronts to the case $d \geq 2$ for i.i.d. random media. We show a wave front propagation speed is attained for the discrete-space (lattice) KPP using a large deviation approach.

4 We construct a nearest-neighbor process ${S_n}$ on Z that is less predictable than simple random walk, in the sense that given the process until time $n$, the conditional probability that $S_{n+k} = x$ is uniformly bounded by $Ck^{-\infty}$ for some $\alpha > 1/2$. From this process, we obtain a probability measure $\mu$ on oriented paths in $\mathbf{Z}^3$ such that the number of intersections of two paths, chosen independently according to $\mu$, has an exponential tail. (For $d \geq 4$, the uniform measure on oriented paths from the origin in $\mathbf{Z}^d$ has this property.) We show that on any graph where such a measure on paths exists, oriented percolation clusters are transient if the retention parameter $p$ is close enough to 1. This yields an extension of a theorem of Grimmett, Kesten and Zhang, who proved that supercritical percolation clusters in $\mathbf{Z}^d$ are transient for all $d \geq 3$.

A few years ago, Grimmett, Kesten and Zhang proved that for supercritical bond percolation on $\mathbf{Z}^3$, simple random walk on the infinite cluster is a.s. transient. We generalize this result to a class of wedges in $\mathbf{Z}^3$ including, for any $\varepsilon \epsilon (0, 1)$, the wedge $\mathscr{W}_{\varepsilon} = {(x, y, z) \epsilon \mathbf{Z}^3: x \geq 0, |z| \leq x^{\varepsilon}}$ which can be thought of as representing a $(2 + \varepsilon)$-dimensional lattice. Our proof builds on recent work of Benjamini, Pemantle and Peres, and involves the construction of finite-energy flows using nearest-neighbor walks on Z with low predictability profile. Along the way, we obtain some new results on attainable decay rates for predictability profiles of nearest-neighbor walks.

We derive the limiting shape for the following model of first-passage bond percolation on the two-dimensional integer lattice: the percolation is directed in the sense that admissible paths are nondecreasing in both coordinate directions. The passage times of horizontal bonds are Bernoulli distributed, while the passage times of vertical bonds are all equal to a deterministic constant. To analyze the percolation model, we couple it with a one-dimensional interacting particle system. This particle process has nonlocal dynamics in the sense that the movement of any given particle can be influenced by far-away particles. We prove a law of large numbers for a tagged particle in this process, and the shape result for the percolation is obtained as a corollary.

We provide a new proof, substantially simpler than Zhang’s original one, that for the contact process on homogeneous trees, local survival implies complete convergence.

We study a one-dimensional particle system in which particles travel deterministically in between stochastic collisions. As the total number of particles tends to infinity, the empirical density converges to a solution of a discrete Boltzmann equation. We establish the large deviation principle for the convergence with a rate function that is given by a variational formula. Some of the properties of the rate function are discussed and a nonvariational expression for the rate function is given.

Let $X$ be a symmetric Lévy process in $\mathbb{R}^d, d = 2, 3$. We assume that $X$ has independent $\alpha_j$-stable components, $1 < \alpha_d \leq \dots \leq \alpha_1 < 2$ (a process with stable components, by Pruitt and Taylor), or more generally that $X$ is $d$-dimensionally self-similar with similarity exponents $H_j, H_j = 1/\alpha_j$ (a dilation-stable process, by Kunita). Let a given integer $k \geq 2$ be such that $k(H - 1) < H, H = \Sigma_{j=1}^d H_j$. We prove that the set of $k$-multiple points $E_k$ is almost surely of Hausdorff dimension

$$\dim E_k = \min (\frac{k - (k -1)H}{H_1}, d - \frac{k(H - 1)}{H_d})$$.

In the stable components case, the above formula was proved by Hendricks for $d = 2$ and was suspected by him for $d = 3$.

For an i.i.d. sequence of random variables with a semiexponential distribution, we give a functional form of the Erdös–Renyi law for partial sums. In contrast to the classical case, that is, the case where the random variables have exponential moments of all orders, the set of limit points is not a subset of the continuous functions. This reflects the bigger influence of extreme values. The proof is based on a large deviation principle for the trajectories of the corresponding random walk. The normalization in this large deviation principle differs from the usual normalization and depends on the tail of the distribution. In the same way, we prove a functional limit law for moving averages.

The $p$-variation of a function $f$ is the supremum of the sums of the $p$th powers of absolute increments of f over nonoverlapping intervals. Let $F$ be a continuous probability distribution function. Dudley has shown that the $p$-variation of the empirical process is bounded in probability as $n \to \infty$ if and only if $p > 2$, and for $1 \leq p \leq 2$, the $p$-variation of the empirical process is at least $n^{1-p/2}$ and is at most of the order $n^{1-p/2}(\log \log n)^{p/2}$ in probability. In this paper, we prove that the exact order of the 2-variation of the empirical process is $\log \log n$ in probability, and for $1 \leq p < 2$, the $p$-variation of the empirical process is of exact order $n^{1-p/2}$ in expectation and almost surely.

Let $S_j := X_1 + X_2 + \dots + X_j$. Then the p-variation of the partial sum process for ${X_1, X_2, \dots, X_n}$ is defined as that of f on $(0, n]$, where $f(t) = S_j$ for $j - 1 < t \leq j, j = 1, 2, \dots, n$. Bretagnolle has shown that the expectation of the $p$-variation for independent centered random variables $X_i$ with bounded $p$th moments is of order $n$ for $1 \leq p < 2$. We prove that for $p = 2$, the 2-variation of the partial sum process of i.i.d. centered nonconstant random variables with finite $2 + \delta$ moment for some $\delta > 0$ is of exact order $n \log \log n$ in probability.

The purpose of this paper is to prove some convergence theorems of closed and convex set valued sub- and supermartingales in the Kuratowski–Mosco sense. To get submartingale convergence theorems, we give sufficient conditions for the Kudo–Aumann integral and Hiai–Umegaki conditional expectation to be closed both for compact convex set valued random variables and for closed convex set valued random variables. We also give an example of a bounded closed convex set valued random variable whose Kudo–Aumann integral is not closed.

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