The existence of stochastic flows in $L^2$ -spaces is proved for a stochastic reaction ] diffusion equation of second order in a bounded domain.

We analyze a new class of one-dimensional interacting particle systems featuring random boundaries with a random motion that is coupled to the local particle density. We show that the hydrodynamic limiting behavior in these systems corresponds to the solution of an appropriate. Stefan (free-boundary) equation and describe some applications of these results.

We consider totally asymmetric attractive zero-range processes with bounded jump rates on Z. In order to obtain a lower bound for the large deviations from the hydrodynamical limit of the empirical measure, we perturb the process in two ways. We first choose a finite number of sites and slow down the jump rate at these sites. We prove a hydrodynamical limit for this perturbed process and show the appearance of Dirac measures on the sites where the rates are slowed down. The second type of perturbation consists of choosing a finite number of particles and making them jump at a slower rate. In these cases the hydrodynamical limit is described by nonentropy weak solutions of quasilinear first-order hyperbolic equations. These two results prove that the large deviations for asymmetric processes with bounded jump rates are of order at least $e^{-CN}$. All these results can be translated to the context of totally asymmetric simple exclusion processes where a finite number of particles or a finite number of holes jump at a slower rate.

A population density process is constructed using approximately $N\imath$ particles performing rate $N^2$ random walks between N cells distributed on the unit interval. Particles give birth or die within cells, and particle death rates are a function of the occupied cell population. With suitable scaling, two possible limiting stochastic partial differential equations are obtained. Both are nonlinear perturbations of the equation satisfied by the density process of super Brownian motion.

We study a system of interacting diffusions $$\begin{eqnarray}dx_\xi(t)&=&\sum_{\zeta\in\Xi}a(\xi,\zeta)(x_\zeta(t)-x_\xi(t))dt \\ && + \sqrt{g(x_\xi(t))} dW_\xi(t) \qquad (\xi\in\Xi),\end{eqnarray}$$ indexed by the hierarchical group $\Xi$, as a genealogical two genotype model [where $x _\xi(t)$ denotes the frequency of, say, type A] with hierarchically determined degrees of relationship between colonies. In the case of short interaction range it is known that the system clusters. That is, locally one genotype dies out. We focus on the description of the different regimes of cluster growth which is shown to depend on the interaction kernel $a(\dot\quad,\dot\quad)$ via its recurrent potential kernel. One of these regimes will be further investigated by mean-field methods. For general interaction range we shall also relate the behavior of large finite systems, indexed by finite subsets $\Xi_n$ of, $\Xi$ to that of the infinite system. On the way we will establish relations between hitting times of random walks and their potentials.

The Fleming-Viot measure-valued diffusion arises as the infinite population limit of various discrete genetic models with general type space. The paper gives a countable construction of the process as the empirical measure carried by a certain interactive particle system. This explicit representation facilitates the study of various properties of the Fleming-Viot process. The construction also carries versions of the familiar genealogical processes from population genetics, in particular, Kingman's coalescent, thus unifying the genealogical and measure-valued approaches to the subject.

Nonlinear martingale problems in the McKean-Vlasov sense for superprocesses are studied. The stochastic calculus on historical trees is used in order to show that there is a unique solution of the nonlinear martingale problems under Lipschitz conditions on the coefficients.

Let $\{S_n, n \geq 0\}$ be a centered d-dimensional random walk $(d \geq 3)$ and consider the so-called future infima process $J_n \stackrel{\mathrm{df}}{=}\inf _{k \geq n} \|S_k\|$. This paper is concerned with obtaining precise integral criteria for a function to be in the Lévy upper class of J. This solves an old problem of Erdös and Taylor, who posed the problem for the simple symmetric random walk on $\mathbb{Z}^d, d \geq 3$. These results are obtained by a careful analysis of the future infima of transient Bessel processes and using strong approximations. Our results belong to a class of Ciesielski-Taylor theorems which relate d-and $(d-2)$-dimensional Bessel processes.

We study low local extremes of Gaussian random fields with values in a separable Hilbert space and constant variance. Our results are sharp for certain stationary processes on the line and for these processes we also prove global limits.

Consider the first exit time and position from small geodesic balls for Brownian motion on Riemannian manifolds. We establish a smooth Besselization technique and calculate the asymptotic expansion for the joint distributions by a purely probabilistic approach.

It is shown here how to extend the spectral characterization of the strong law of large numbers for weakly stationary processes to certain singular averages. For instance, letting $\{X_t, t \in R^3\}$ be a weakly stationary field, $\{X_t\}$ satisfies the usual SLLN (by averaging over balls) if and only if the averages of $\{X_t\}$ over spheres of increasing radii converge pointwise. The same result in two dimensions is false. This spectral approach also provides a necessary and sufficient condition for the a.s. convergence of some series of stationary variables.

There is a simple inequality by Pinsker between variational distance and informational divergence of probability measures defined on arbitrary probability spaces. We shall consider probability measures on sequences taken from countable alphabets, and derive, from Pinsker's inequality, bounds on the $\bar{d}$-distance by informational divergence. Such bounds can be used to prove the "concentration of measure" phenomenon for some nonproduct distributions.

The celebrated theorem of Halmos and Savage implies that if M is a set of $\mathbb{P}$-absolutely continuous probability measures Q on $(\Omega, \mathscr{F}, \mathbb{P})$ such that each $A \in \mathscr{F}, \mathbb{P}(A) > 0$ is charged by some $Q\in M$, that is, $Q(A) > 0$ (where the choice of Q depends on the set A), then -- provided M is closed under countable convex combinations -- we can find $Q_0 \in M$ with full support; that is, $\mathbb{P}(A) > 0$ implies $Q_0(A) > 0 $. We show a quantitative version: if we assume that, for $\varepsilon > 0$ and $\delta > 0$ fixed, $\mathbb{P}(A)> \varepsilon$ implies that there is $Q \in M$ and $Q(A) > \delta$, then there is $Q_0 \in M$ such that $\mathbb{P}(A) >4 \varepsilon$ implies $Q_0(A)>\varepsilon^2 \delta/2$. This version of the Halmos-Savage theorem also allows a "dualization" which we also prove in a quantitative and qualitative version. We give applications to asymptoic problems arising in mathematical finance and pertaining to the relation of the concept of "no arbitrage" and the existence of equivalent martingale measures for a sequence of stochastic processes.

Let $(\mathscr{F}_t)_{t \geq 0}$ be the filtration of a Brownian motion $(B(t))_{t \geq 0}$ on $(\Omega,\mathscr{F},P)$. An example is given of a measure $Q \sim P$ (in the sense of absolute continuity) for which $(\mathscr{F}_t)_{t \geq 0}$ is not the filtration of any Brownian motion on $(\Omega,\mathscr{F},Q)$. This settles a 15-year-old question.

Sharpening the main result of the preceding paper, it is shown that if, $B_t,0 \leq t < \infty$ is a standard Brownian motion on $(\Omega,\mathscr{F},P)$, then for any $\varepsilon > 0$ there is a probability measure $Q$ with $(1 - \varepsilon)P \leq Q \leq (1= \varepsilon)P$ such that the filtration of B cannot be generated by any Brownian motion on $(\Omega,\mathscr{F},Q)$.

A consequence of the preceding two papers is this. Let $\{\mathscr{A}_t: 0 \leq t < \infty\}$ be the filtration of a stochastic process on $(\Omega, \mathscr{A},P)$. Under a mild assumption on the process, there exist, for any $\varepsilon > 0$, uncountably many probability measures $Q_\alpha$ with $(1 - \varepsilon) P \leq Q_\alpha \leq (1+ \varepsilon)P$ so that no two of the filtrations $(\Omega, (\mathscr{A}_t)_{o \leq t}, Q_\alpha)$ and $(\Omega (\mathscr{A}_t)_{o\leq t}, Q_\beta), \alpha \not= \beta$, can be generated by equivalent stochastic processes.

In this paper we consider $\psi$-irreducible Markov processes evolving in discrete or continuous time on a general state space. We develop a Liapounov function criterion that permits one to obtain explicit bounds on the solution to the Poisson equation and, in particular, obtain conditions under which the solution is square integrable.

These results are applied to obtain sufficient conditions that guarantee the validity of a functional central limit theorem for the Markov process. As a second consequence of the bounds obtained, a perturbation theory for Markov processes is developed which gives conditions under which both the solution to the Poisson equation and the invariant probability for the process are continuous functions of its transition kernel. The techniques are illustrated with applications to queueing theory and autoregressive processes.

In this paper we get some sufficient conditions for the finiteness or nonfiniteness of the passage-time moments for nonnegative discrete parameter processes. The developed criteria are closely connected with the well-known results of Foster for the ergodicity of Markov chains and are given in terms of sub(super)martingales. Then, as an application of the obtained results, we get explicit conditions for the finiteness or nonfiniteness of passage-time moments for reflected random walks in a quadrant with zero drift in the interior.

A rather complicated condition is shown to be necessary and sufficient for a Lévy process to have points of increase. A much simpler condition is then shown to be sufficient in the general case, and necessary under certain regularity conditions. The approach used here also gives a unified proof of results for certain special classes of Lévy processes, which have previously been obtained by Bertoin.

There has been substantial interest in the indices $0 \leq \beta'' \leq \beta' \leq \beta \leq 2$, defined by Blumenthal and Getoor, determined by a general Lévy process in $\mathbf{R}^d$. Pruitt defined an index $\gamma$ which determines the covering dimension and Taylor showed that an index $\gamma'$, first considered by Hendricks, determines the packing dimension for the trajectory. In the present paper we prove that $$\frac{\beta}{2} \le \gamma' \le \min(\beta, d),$$ and give examples to show that the whole range is attainable. However, we cannot completely determine the set of values of $(\gamma, \gamma', \beta)$ which can be attained as indices of some Lévy process.

This paper examines random walks on a finite group G and finds upper bounds on how long it takes typical random walks supported on $(\log|G|)^a$ elements to get close to uniformly distributed on G. For certain groups, a cutoff phenomenon is shown to exist for these typical random walks. A variation of the upper bound lemma of Diaconis and Shahshahani and some counting arguments related to a group equation are used to get the upper bound. A further example which uses this variation is discussed.

We estimate the expected mixing time of a random walk on a finite group supported by a random polylogarithmic set of elements. Following the spectral approach of Broder and Shamir, we present an alternative proof of the Dou-Hildebrand estimate and show that it holds almost surely. Good bounds on diameters follow from these results.

Let $(S_n )_{n \in \mathbb {N}_0}$ be a random walk on the integers having bounded steps. The self-repellent (resp., self-avoiding) walk is a sequence of transformed path measures which discourage (resp., forbid) self-intersections. This is used as a model for polymers. Previously, we proved a law of large numbers; that is, we showed the convergence of $|S_n | /n$ toward a positive number $\theta$ under the polymer measure. The present paper proves a classical central limit theorem for the self-repellent and self-avoiding walks; that is, we prove the asymptotic normality of $(S_n - \theta_n) / \sqrt{n}$ for large n. The proof refines and continues results and techniques developed previously.

We prove large deviation estimates at the correct order for the graph distance of two sites lying in the same cluster of an independent percolation process. We improve earlier results of Gärtner and Molchanov and Grimmett and Marstrand and answer affirmatively a conjecture of Kozlov.