We consider a random walk on the $d$-dimensional lattice and investigate the asymptotic probability of the walk avoiding a "disaster" (points put down according to a regular Poisson process on space-time). We show that, given the Poisson process points, almost surely, the chance of surviving to time $t$ is like $e^{-\alpha \log (\frac1k) t } $, as $t$ tends to infinity if $k$, the jump rate of the random walk, is small.

We analyze the drunkard's walk on the unit sphere with step size $\theta$ and show that the walk converges in order $C/\sin^2(\theta)$ steps in the discrepancy metric ($C$ a constant). This is an application of techniques we develop for bounding the discrepancy of random walks on Gelfand pairs generated by bi-invariant measures. In such cases, Fourier analysis on the acting group admits tractable computations involving spherical functions. We advocate the use of discrepancy as a metric on probabilities for state spaces with isometric group actions.

We prove that the number of maximal points in a random sample taken uniformly and independently from a convex polygon is asymptotically normal in the sense of convergence in distribution. Many new results for other planar regions are also derived. In particular, precise Poisson approximation results are given for the number of maxima in regions bounded above by a nondecreasing curve.

Let ${\Bbb Z}^2_{cp}$ be the close-packed graph of $\Bbb Z^2$, that is, the graph obtained by adding to each face of $\Bbb Z^2$ its diagonal edges. We consider site percolation on $\Bbb Z^2_{cp}$, namely, for each $v$ we choose $X(v) = 1$ or 0 with probability $p$ or $1-p$, respectively, independently for all vertices $v$ of $\Bbb Z^2_{cp}$. We say that a word $(\xi_1, \xi_2,\dots) \in \{0,1\}^{\Bbb N}$ is seen in the percolation configuration if there exists a selfavoiding path $(v_1, v_2, \dots)$ on $\Bbb Z^2_{cp}$ with $X(v_i) = \xi_i, i \ge 1$. $p_c(\Bbb Z^2, \text{site})$ denotes the critical probability for site-percolation on $\Bbb Z^2$. We prove that for each fixed $p \in \big (1- p_c(\Bbb Z^2, \text{site}), p_c(\Bbb Z^2, \text{site})\big )$, with probability 1 all words are seen. We also show that for some constants $C_i \gt 0$ there is a probability of at least $C_1$ that all words of length $C_0n^2$ are seen along a path which starts at a neighbor of the origin and is contained in the square $[-n,n]^2$.

If $\mu$ is a probability measure on the set of suitable weak solutions of the 3D Navier-Stokes equations, invariant for the time-shift, with finite mean dissipation rate, then at every time $t$ the set of singular points is empty $\mu$-a.s. The existence of a measure $\mu$ with the previous properties is also proved; it may describe a turbulent asymptotic regime.

We consider deterministic and disordered frustrated systems in which we can show some strict inequalities with respect to related ferromagnetic systems. A case particularly interesting is the Edwards-Anderson spin-glass model in which it is possible to determine a region of uniqueness of the Gibbs measure, which is strictly larger than the region of uniqueness for the related ferromagnetic system. We analyze also deterministic systems with $|J_b| \in [J_A, J_B]$ where $0 \lt J_A \leq J_B \lt \infty$, for which we prove strict inequality for the critical points of the related FK model. The results are obtained for the Ising models but some extensions to Potts models are possible.

A time-space harmonic polynomial for a continuous-time process $X=\{X_t : t \ge 0\} $ is a two-variable polynomial $ P $ such that $ \{ P(t,X_t) : t \ge 0 \} $ is a martingale for the natural filtration of $ X $. Motivated by Lévy's characterisation of Brownian motion and Watanabe's characterisation of the Poisson process, we look for classes of processes with reasonably general path properties in which a characterisation of those members whose laws are determined by a finite number of such polynomials is available. We exhibit two classes of processes, the first containing the Lévy processes, and the second a more general class of additive processes, with this property and describe the respective characterisations.

We give a principle of large deviations for a generalized version of the strong central limit theorem. This generalized version deals with martingale additive functionals of a recurrent Markov process.

We study the asymptotics as $t \to 0$ of the transition density of a class of $\mu$-symmetric diffusions in the case when the measure $\mu$ has a multi-fractal structure. These diffusions include singular time changes of Brownian motion on the unit cube.

Consider the radial projection onto the unit sphere of the path a $d$-dimensional Brownian motion $W$, started at the center of the sphere and run for unit time. Given the occupation measure $\mu$ of this projected path, what can be said about the terminal point $W(1)$, or about the range of the original path? In any dimension, for each Borel set $A$ in $S^{d-1}$, the conditional probability that the projection of $W(1)$ is in $A$ given $\mu(A)$ is just $\mu(A)$. Nevertheless, in dimension $d \ge 3$, both the range and the terminal point of $W$ can be recovered with probability 1 from $\mu$. In particular, for $d \ge 3$ the conditional law of the projection of $W(1)$ given $\mu$ is not $\mu$. In dimension 2 we conjecture that the projection of $W(1)$ cannot be recovered almost surely from $\mu$, and show that the conditional law of the projection of $W(1)$ given $\mu$ is not $mu$.

We develop a method for analyzing the mixing times for a quite general class of Markov chains on the complete monomial group $G \wr S_n$ and a quite general class of Markov chains on the homogeneous space $(G\wr S_n) / (S_r\times S_{n-r})$. We derive an exact formula for the $L^2$ distance in terms of the $L^2$ distances to uniformity for closely related random walks on the symmetric groups $S_j$ for $1 \leq j \leq n$ or for closely related Markov chains on the homogeneous spaces $S_{i+j}/ (S_i~\times~S_j)$ for various values of $i$ and $j$, respectively. Our results are consistent with those previously known, but our method is considerably simpler and more general.

We extend Krylov's $L_p$-solvability theory to the Cauchy problem for systems of parabolic stochastic partial differential equations. Some additional integrability and regularity properties are also presented.

We study boundary conditions for a stochastic pseudo processes corresponding to the biharmonic operator. The biharmonic pseudo process (BPP for short). is composed, in a sense, of two different particles, a monopole and a dipole. We show how an initial-boundary problems for a 4-th order parabolic differential equation can be represented by BPP with various boundary conditions for the two particles: killing, reflection and stopping.

We study here the fragmentation processes that can be derived from Lévy processes with no positive jumps in the same manner as in the case of a Brownian motion (cf. Bertoin [4]). One of our motivations is that such a representation of fragmentation processes by excursion-type functions induces a particular order on the fragments which is closely related to the additivity of the coalescent kernel. We identify the fragmentation processes obtained this way as a mixing of time-reversed extremal additive coalescents by analogy with the work of Aldous and Pitman [2], and we make its semigroup explicit.

Two different problems are studied:

• 1. For an infinite locally finite connected graph $G$, let $p_c(G)$ be the critical value for the existence of an infinite cluster in iid bond percolation on $G$ and let $P_c = \sup\{p_c(G): G \text{ transitive }, p_c(G) \lt 1\}$. Is $P_c \lt 1$?

• 2. Let $G$ be transitive with $p_c(G) \lt 1$, take $p \in [0,1]$ and let $X$ and $Y$ be iid bond percolations on $G$ with retention parameters $(1+p)/2$ and $(1-p)/2$ respectively. Is there a $q \lt 1$ such that $p \gt q$ implies that for any monotone coupling $(X',Y')$ of $X$ and $Y$ the edges for which $X'$ and $Y'$ disagree form infinite connected component(s) with positive probability? Let $p_d(G)$ be the infimum of such $q$'s (including $q=1$) and let $P_d = \sup\{p_d(G): G \text{ transitive }, p_c(G) \lt 1\}$. Is the stronger statement $P_d \lt 1$ true? On the other hand: Is it always true that $p_d(G) \gt p_c (G)$?

It is shown that if one restricts attention to biregular planar graphs then these two problems can be treated in a similar way and all the above questions are positively answered. We also give examples to show that if one drops the assumption of transitivity, then the answer to the above two questions is no. Furthermore it is shown that for any bounded-degree bipartite graph $G$ with $p_c(G) \lt 1$ one has $p_c(G) \lt p_d(G)$. Problem (2) arises naturally from [6] where an example is given of a coupling of the distinct plus- and minus measures for the Ising model on a quasi-transitive graph at super-critical inverse temperature. We give an example of such a coupling on the $r$-regular tree, ${\bf T}_r$, for $r \gt 1$.

The asymptotic behavior of a general class of discrete generation interacting particle systems is discussed. We provide $L_p$-mean error estimates for their empirical measure on path space and present sufficient conditions for uniform convergence of the particle density profiles with respect to the time parameter. Several examples including mean field particle models, genetic schemes and McKean's Maxwellian gases will also be given. In the context of Feynman-Kac type limiting distributions we also prove central limit theorems and we start a variance comparison for two generic particle approximating models.

Stationary distributions of Markov processes can typically be characterized as probability measures that annihilate the generator in the sense that $int_EAfd\mu =0$ for $fin {cal D}(A)$; that is, for each such $\mu$, there exists a stationary solution of the martingale problem for $A$ with marginal distribution $\mu$. This result is extended to models corresponding to martingale problems that include absolutely continuous and singular (with respect to time) components and controls. Analogous results for the forward equation follow as a corollary.

We consider domains $D$ of $R^d$, $d\ge 2$ with the property that there is a wedge $V\subset R^d$ which is left invariant under all tangential projections at smooth portions of $\partial D$. It is shown that the difference between two solutions of the Skorokhod equation in $D$ with normal reflection, driven by the same Brownian motion, remains in $V$ if it is initially in $V$. The heat equation on $D$ with Neumann boundary conditions is considered next. It is shown that the cone of elements $u$ of $L^2(D)$ satisfying $u(x)-u(y)\ge0$ whenever $x-y\in V$ is left invariant by the corresponding heat semigroup. Positivity considerations identify an eigenfunction corresponding to the second Neumann eigenvalue as an element of this cone. For $d=2$ and under further assumptions, especially convexity of the domain, this eigenvalue is simple.

We are interested in the component intervals of the complements of a monotone sequence $R_n \subseteq \dots \subseteq R_1$ of regenerative sets, for two natural embeddings. One is based on Bochner's subordination, and one on the intersection of independent regenerative sets. For each scheme, we study the joint law of the so-called ages and residual lifetimes.

The model studied concerns a simple first-order hyperbolic system. The solutions in which one is most interested have discontinuities which persist for all time, and therefore need to be interpreted as weak solutions. We demonstrate existence and uniqueness for such weak solutions, identifying a canonical `exact' solution which is everywhere defined. The direct method used is guided by the theory of measure-valued diffusions. The method is more effective than the method of characteristics, and has the advantage that it leads immediately to the McKean representation without recourse to Itô's formula.

We then conduct computer studies of our model, both by integration schemes (which do use characteristics) and by `random simulation'.

The small diffusion and fast dying out asymptotics is calculated for nonlinear equations of a class of superprocesses on manifolds, and the corresponding logarithmic limit of the solution is shown to be given by a solution of a certain problem of calculus of variations with a non-additive (and non-integral) functional.

This paper proves upper and lower off-diagonal, sub-Gaussian transition probabilities estimates for strongly recurrent random walks under sufficient and necessary conditions. Several equivalent conditions are given showing their particular role and influence on the connection between the sub-Gaussian estimates, parabolic and elliptic Harnack inequality.

Suppose that $G_j$ is a sequence of finite connected planar graphs, and in each $G_j$ a special vertex, called the root, is chosen randomly-uniformly. We introduce the notion of a distributional limit $G$ of such graphs. Assume that the vertex degrees of the vertices in $G_j$ are bounded, and the bound does not depend on $j$. Then after passing to a subsequence, the limit exists, and is a random rooted graph $G$. We prove that with probability one $G$ is recurrent. The proof involves the Circle Packing Theorem. The motivation for this work comes from the theory of random spherical triangulations.

A class of linear degenerate second-order parabolic equations is considered in arbitrary domains. It is shown that these equations are solvable using special weighted Sobolev spaces in essentially the same way as the non-degenerate equations in $R^d$ are solved using the usual Sobolev spaces. The main advantages of this Sobolev-space approach are less restrictive conditions on the coefficients of the equation and near-optimal space-time regularity of the solution. Unlike previous works on degenerate equations, the results cover both classical and distribution solutions and allow the domain to be bounded or unbounded without any smoothness assumptions about the boundary. An application to nonlinear filtering of diffusion processes is discussed.

We construct a class of superprocesses by taking the high density limit of a sequence of interacting-branching particle systems. The spatial motion of the superprocess is determined by a system of interacting diffusions, the branching density is given by an arbitrary bounded non-negative Borel function, and the superprocess is characterized by a martingale problem as a diffusion process with state space $M({\bf R})$, improving and extending considerably the construction of Wang (1997, 1998). It is then proved in a special case that a suitable rescaled process of the superprocess converges to the usual super Brownian motion. An extension to measure-valued branching catalysts is also discussed.

The Fractional Brownian Motion can be extended to complex values of the parameter $\alpha $ for $\Re\alpha \gt {1\over 2}$. This is a useful tool. Indeed, the obtained process depends holomorphically on the parameter, so that many formulas, as Itô formula, can be extended by analytic continuation. For large values of $\Re\alpha $, the stochastic calculus reduces to a deterministic one, so that formulas are very easy to prove. Hence they hold by analytic continuation for $\Re\alpha \le 1$, containing the classical case $\alpha =1$.