In this paper we give two examples of evanescent Markov chains which exhibit unusual behaviour on conditioning to survive for large times. In the first example we show that the conditioned processes converge vaguely in the discrete topology to a limit with a finite lifetime, but converge weakly in the Martin topology to a non-Markovian limit. In the second example, although the family of conditioned laws are tight in the Martin topology, they possess multiple limit points so that weak convergence fails altogether.

The probability that the cluster of the origin in critical site percolation on the triangular grid has diameter larger than $R$ is proved to decay like $R$ to the power $5/48$ as $R$ goes to infinity.

For nonlinear functions $f$ of a random vector $Y$, $E[f(Y)]$ and $f(E[Y])$ usually differ. Consequently the mathematical expectation of $Y$ is not intrinsic: when we change coordinate systems, it is not invariant.This article is about a fundamental and hitherto neglected property of random vectors of the form $Y = f(X(t))$, where $X(t)$ is the value at time $t$ of a diffusion process $X$: namely that there exists a measure of location, called the ``intrinsic location parameter'' (ILP), which coincides with mathematical expectation only in special cases, and which is invariant under change of coordinate systems. The construction uses martingales with respect to the intrinsic geometry of diffusion processes, and the heat flow of harmonic mappings. We compute formulas which could be useful to statisticians, engineers, and others who use diffusion process models; these have immediate application, discussed in a separate article, to the construction of an intrinsic nonlinear analog to the Kalman Filter. We present here a numerical simulation of a nonlinear SDE, showing how well the ILP formula tracks the mean of the SDE for a Euclidean geometry.

A Large Deviation Principle (LDP) is proved for the family $(1/n)\sum_1^n f(x_i^n) Z_i$ where $(1/n)\sum_1^n \delta_{x_i^n}$ converges weakly to a probability measure on $R$ and $(Z_i)_{i\in N}$ are $R^d$-valued independent and identically distributed random variables having some exponential moments, i.e., $$E e^{a |Z|} \lt \infty$$ for some $0 \lt a \lt \infty$. The main improvement of this work is the relaxation of the steepness assumption concerning the cumulant generating function of the variables $(Z_i)_{i \in N}$. In fact, Gärtner-Ellis' theorem is no longer available in this situation. As an application, we derive a LDP for the family of empirical measures $(1/n) \sum_1^n Z_i \delta_{x_i^n}$. These measures are of interest in estimation theory (see Gamboa et al., Csiszar et al.), gas theory (see Ellis et al., van den Berg et al.), etc. We also derive LDPs for empirical processes in the spirit of Mogul'skii's theorem. Various examples illustrate the scope of our results.

We show that the function $h(x)=\prod_{i \lt j}(x_j-x_i)$ is harmonic for any random walk in $R^k$ with exchangeable increments, provided the required moments exist. For the subclass of random walks which can only exit the Weyl chamber $W=\{x\colon x_1 \lt x_2 \lt \cdots \lt x_k\}$ onto a point where $h$ vanishes, we define the corresponding Doob $h$-transform. For certain special cases, we show that the marginal distribution of the conditioned process at a fixed time is given by a familiar discrete orthogonal polynomial ensemble. These include the Krawtchouk and Charlier ensembles, where the underlying walks are binomial and Poisson, respectively. We refer to the corresponding conditioned processes in these cases as the Krawtchouk and Charlier processes. In [O'Connell and Yor (2001b)], a representation was obtained for the Charlier process by considering a sequence of $M/M/1$ queues in tandem. We present the analogue of this representation theorem for the Krawtchouk process, by considering a sequence of discrete-time $M/M/1$ queues in tandem. We also present related results for random walks on the circle, and relate a system of non-colliding walks in this case to the discrete analogue of the circular unitary ensemble (CUE).

We give sharp rates of convergence for a natural Markov chain on the space of phylogenetic trees and dually for the natural random walk on the set of perfect matchings in the complete graph on $2n$ vertices. Roughly, the results show that $(1/2) n \log n$ steps are necessary and suffice to achieve randomness. The proof depends on the representation theory of the symmetric group and a bijection between trees and matchings.

We study the $d$-dimensional branching random walk for $d \gt 2$. This process has extremal equilibria for every intensity. We are interested in the large space scale and large space-time scale behavior of the equilibrium state. We show that the fluctuations of space and space-time averages with a non-classical scaling are Gaussian in the limit. For this purpose we use the historical process, which allows a family decomposition. To control the distribution of the families we use the concept of canonical measures and Palm distributions.

We consider Markov chains on the space of (countable) partitions of the interval $[0,1]$, obtained first by size biased sampling twice (allowing repetitions) and then merging the parts with probability $\beta_m$ (if the sampled parts are distinct) or splitting the part with probability $\beta_s$, according to a law $\sigma$ (if the same part was sampled twice). We characterize invariant probability measures for such chains. In particular, if $\sigma$ is the uniform measure, then the Poisson-Dirichlet law is an invariant probability measure, and it is unique within a suitably defined class of "analytic" invariant measures. We also derive transience and recurrence criteria for these chains.

Consider a uniformly chosen element $X_n$ of the $n$-fold wreath product $\Gamma_n = G \wr G \wr \cdots \wr G$, where $G$ is a finite permutation group acting transitively on some set of size $s$. The eigenvalues of $X_n$ in the natural $s^n$-dimensional permutation representation (the composition representation) are investigated by considering the random measure $\Xi_n$ on the unit circle that assigns mass $1$ to each eigenvalue. It is shown that if $f$ is a trigonometric polynomial, then $\lim_{n \rightarrow \infty} P\{\int f d\Xi_n \ne s^n \int f d\lambda\}=0$, where $\lambda$ is normalised Lebesgue measure on the unit circle. In particular, $s^{-n} \Xi_n$ converges weakly in probability to $\lambda$ as $n \rightarrow \infty$. For a large class of test functions $f$ with non-terminating Fourier expansions, it is shown that there exists a constant $c$ and a non-zero random variable $W$ (both depending on $f$) such that $c^{-n} \int f d\Xi_n$ converges in distribution as $n \rightarrow \infty$ to $W$. These results have applications to Sylow $p$-groups of symmetric groups and autmorphism groups of regular rooted trees.

We consider Funaki's model of a random string taking values in $\mathbf{R}^d$. It is specified by the following stochastic PDE, \[ \frac{\partial u(x)}{\partial t}=\frac{\partial^2 u(x)}{\partial x^2} +\dot{W}. \] where $\dot{W}=\dot{W}(x,t)$ is two-parameter white noise, also taking values in $\mathbf{R}^d$. We find the dimensions in which the string hits points, and in which it has double points of various types. We also study the question of recurrence and transience.

We study the behavior of shocks in the asymmetric simple exclusion process on $Z$ whose initial distribution is a product measure with a finite number of shocks. We prove that if the particle hopping rates of this process are in a particular relation with the densities of the initial measure then the distribution of this process at any time is a linear combination of shock measures of the structure similar to that of the initial distribution. The structure of this linear combination allows us to interpret this result by saying that the shocks of the initial distribution perform continuous time random walks on $Z$ interacting by the exclusion rule. We give explicit expressions for the hopping rates of these random walks. The result is derived with a help of quantum algebra technique. We made the presentation self-contained for the benefit of readers not acquainted with this approach, but interested in applying it in the study of interacting particle systems.

We investigate the two-point stochastic boundary-value problem on $[0,1]$: \begin{equation}\label{1} \begin{split} U'' &= f(U)\dot W + g(U,U')\\ U(0) &= \xi\\ U(1)&= \eta. \end{split} \tag{1} \end{equation} where $\dot W$ is a white noise on $[0,1]$, $\xi$ and $\eta$ are random variables, and $f$ and $g$ are continuous real-valued functions. This is the stochastic analogue of the deterministic two point boundary-value problem, which is a classical example of bifurcation. We find that if $f$ and $g$ are affine, there is no bifurcation: for any r.v. $\xi$ and $\eta$, (1) has a unique solution a.s. However, as soon as $f$ is non-linear, bifurcation appears. We investigate the question of when there is either no solution whatsoever, a unique solution, or multiple solutions. We give examples to show that all these possibilities can arise. While our results involve conditions on $f$ and $g$, we conjecture that the only case in which there is no bifurcation is when $f$ is affine.

We consider two types of measure-valued branching processes on the lattice $Z^d$. These are on the one hand side a particle system, called branching random walk, and on the other hand its continuous mass analogue, a system of interacting diffusions also called super random walk. It is known that the long-term behavior differs sharply in low and high dimensions: if $d\leq 2$ one gets local extinction, while, for $d\geq 3$, the systems tend to a non-trivial equilibrium. Due to Kallenberg's criterion, local extinction goes along with clumping around a 'typical surviving particle.' This phenomenon is called clustering. A detailed description of the clusters has been given for the corresponding processes on $R^2$ in Klenke (1997). Klenke proved that with the right scaling the mean number of particles over certain blocks are asymptotically jointly distributed like marginals of a system of coupled Feller diffusions, called system of tree indexed Feller diffusions, provided that the initial intensity is appropriately increased to counteract the local extinction. The present paper takes different remedy against the local extinction allowing also for state-dependent branching mechanisms. Instead of increasing the initial intensity, the systems are described under the Palm distribution. It will turn out together with the results in Klenke (1997) that the change to the Palm measure and the multiple scale analysis commute, as $t\to\infty$. The method of proof is based on the fact that the tree indexed systems of the branching processes and of the diffusions in the limit are completely characterized by all their moments. We develop a machinery to describe the space-time moments of the superprocess effectively and explicitly.

The distributions of Wiener functionals of second order are infinitely divisible. An explicit expression of the associated Lévy measures in terms of the eigenvalues of the corresponding Hilbert-Schmidt operators on the Cameron-Martin subspace is presented. In some special cases, a formula for the densities of the distributions is given. As an application of the explicit expression, an exponential decay property of the characteristic functions of the Wiener functionals is discussed. In three typical examples, complete descriptions are given.

A two-type infinite-measure-valued population in $R^2$ is constructed which undergoes diffusion and branching. The system is interactive in that the branching rate of each type is proportional to the local density of the other type. For a collision rate sufficiently small compared with the diffusion rate, the model is constructed as a pair of infinite-measure-valued processes which satisfy a martingale problem involving the collision local time of the solutions. The processes are shown to have densities at fixed times which live on disjoint sets and explode as they approach the interface of the two populations. In the long-term limit (in law), local extinction of one type is shown. Moreover the surviving population is uniform with random intensity. The process constructed is a rescaled limit of the corresponding $Z^2$-lattice model studied by Dawson and Perkins (1998) and resolves the large scale mass-time-space behavior of that model under critical scaling. This part of a trilogy extends results from the finite-measure-valued case, whereas uniqueness questions are again deferred to the third part.

We study a system of simple random walks on graphs, known as frog model. This model can be described as follows: There are active and sleeping particles living on some graph. Each active particle performs a simple random walk with discrete time and at each moment it may disappear with probability $1-p$. When an active particle hits a sleeping particle, the latter becomes active. Phase transition results and asymptotic values for critical parameters are presented for $Z^d$ and regular trees.

Our main object that we call the Poisson snake is a Brownian snake as introduced by Le Gall. This process has values which are trajectories of standard Poisson process stopped at some random finite lifetime with Brownian evolution. We use this Poisson snake to construct a self-similar fragmentation as introduced by Bertoin. A similar representation was given by Aldous and Pitman using the Continuum Random Tree. Whereas their proofs used approximation by discrete models, our representation allows continuous time arguments.

We study the limit of functionals of stochastic processes for which an homogenization result holds. All these functionals involve stochastic integrals. Among them, we consider more particularly the Levy area and those giving the solutions of some SDEs. The main question is to know whether or not the limit of the stochastic integrals is equal to the stochastic integral of the limit of each of its terms. In fact, the answer may be negative, especially in presence of a highly oscillating first-order differential term. This provides us some counterexamples to the theory of good sequence of semimartingales.

We give sufficient conditions for non-explosion and transience of the solution $(x_t, p_t)$ (in dimensions $\geq 3$) to a stochastic Newtonian system of the form $$\begin{cases} dx_t= p_t \, dt , \\ dp_t= -\frac{\partial V(x_t) }{\partial x} \, dt - \frac{ \partial c(x_t) }{ \partial x} \, d\xi_t,\end{cases}$$ where $\{\xi_t\}_{t\geq 0}$ is a $d$-dimensional Lévy process, $d\xi_t$ is an Itô differential and $c\in C^2(\mathbb{R}^d,\mathbb{R}^d)$, $V\in C^2(\mathbb{R}^d,\mathbb{R})$ such that $V\geq 0$.

We study a diffusion process with a molecular diffusion and random Markovian-Gaussian drift for which the usual (spatial) Peclet number is infinite. We introduce a temporal Peclet number and we prove that, under the finiteness of the temporal Peclet number, the laws of diffusions under the diffusive rescaling converge weakly, to the law of a Brownian motion. We also show that the effective diffusivity has a finite, nonzero limit as the molecular diffusion tends to zero.

The purpose of this article is to study a coalescing flow of sticky Brownian motions. Sticky Brownian motion arises as a weak solution of a stochastic differential equation, and the study of the flow reveals the nature of the extra randomness that must be added to the driving Brownian motion. This can be represented in terms of Poissonian marking of the trees associated with the excursions of Brownian motion. We also study the noise, in the sense of Tsirelson, generated by the flow. It is shown that this noise is not generated by any Brownian motion, even though it is predictable.

We consider a class of jump-diffusion processes, constrained to a polyhedral cone $G\subset\mathbb{R}^n$, where the constraint vector field is constant on each face of the boundary. The constraining mechanism corrects for ``attempts'' of the process to jump outside the domain. Under Lipschitz continuity of the Skorohod map $\Gamma$, it is known that there is a cone ${\cal C}$ such that the image $\Gamma\phi$ of a deterministic linear trajectory $\phi$ remains bounded if and only if $\dot\phi\in{\cal C}$. Denoting the generator of a corresponding unconstrained jump-diffusion by $\cal L$, we show that a key condition for the process to admit an invariant probability measure is that for $x\in G$, ${\cal L}\,{\rm id}(x)$ belongs to a compact subset of ${\cal C}^o$.

We prove a self-normalized large deviation principle for sums of Banach space valued functions of a Markov chain. Self-normalization applies to situations for which a full large deviation principle is not available. We follow the lead of Dembo and Shao [DemSha98b] who state partial large deviations principles for independent and identically distributed random sequences.