We consider a stochastic flow driven by a finite-dimensional Brownian motion. We show that almost every realization of such a flow exhibits strong statistical properties such as the exponential convergence of an initial measure to the equilibrium state and the central limit theorem. The proof uses new estimates of the mixing rates of the multi-point motion.

Let $X,X_j,j\in \mathbb{N}$, be independent, identically distributed random variables with probability distribution F. It is shown that Student's statistic of the sample $\{X_j\}_{j=1}^n$ has a limit distribution G such that $G(\{-1,1\})\ne 1$, if and only if: (1) $X$ is in the domain of attraction of a stable law with some exponent $0<\alpha\le 2$; (2) $\E X=0$ if $1<\alpha\le 2;$ (3) if $\alpha=1$, then X is in the domain of attraction of Cauchy's law and Feller's condition holds: $\lim_{n\to\infty}n\E\sin(X/a_n)$ exists and is finite, where $a_n$ is the infimum of all $x>0$ such that $nx^{-2}(1+\int_{(-x,x)} y^2 F \{dy\})\le 1$. If $G(\{-1,1\})=1$, then Student's statistic of the sample $\{X_j\}_{j=1}^n$ has a limit distribution if and only if $\P(|X|>x),x>0$, is a slowly varying function at $+\infty$.

Let X be either the branching diffusion corresponding to the operator $Lu+\beta (u^2-u)$ on $D\subseteq $ $\mathbb{R}^{d}$ [where $\beta (x) \geq 0$ and $\beta\not\equiv 0$ is bounded from above] or the superprocess corresponding to the operator $Lu+\beta u -\alpha u^2$ on $D\subseteq $ $\mathbb{R}^{d}$ (with $\alpha>0$ and $\beta$ is bounded from above but no restriction on its sign). Let $\lambda _{c}$ denote the generalized principal eigenvalue for the operator $L+\beta $ on $D$. We prove the following dichotomy: either $\lambda _{c}\leq 0$ and X exhibits local extinction or $\lambda _{c}> 0$ and there is exponential growth of mass on compacts of D with rate $\lambda _{c}$. For superdiffusions, this completes the local extinction criterion obtained by Pinsky [Ann. Probab. 24 (1996) 237--267] and a recent result on the local growth of mass under a spectral assumption given by Engländer and Turaev [Ann. Probab. 30 (2002) 683--722]. The proofs in the above two papers are based on PDE techniques, however the proofs we offer are probabilistically conceptual. For the most part they are based on "spine'' decompositions or "immortal particle representations'' along with martingale convergence and the law of large numbers. Further they are generic in the sense that they work for both types of processes.

We prove that the heat kernel on the infinite Bernoulli percolation cluster in $\Z^d$ almost surely decays faster than $t^{-d/2}$. We also derive estimates on the mixing time for the random walk confined to a finite box. Our approach is based on local isoperimetric inequalities. Some of the results of this paper were previously announced in the note of Mathieu and Remy [C. R. Acad. Sci. Paris Sér. I Math. 332 (2001) 927--931].

Consider a simple symmetric random walk on the integer lattice $\ZB$. For each n, let $V(n)$ denote a favorite site (or most visited site) of the random walk in the first n steps. A somewhat surprising theorem of Bass and Griffin [Z. Wahrsch. Verw. Gebiete 70 (1985) 417--436] says that V is almost surely transient, thus disproving a previous conjecture of Erdős and Révész [Mathematical Structures--Computational Mathematics--Mathematical Modeling 2 (1984) 152--157]. More precisely, Bass and Griffin proved that almost surely, $\liminf_{n\to \infty} {|V(n)| \over n^{1/2}(\log n)^{-\gamma}}$ equals $0$ if $\gamma<:1$, and is infinity if $\gamma>11$ (eleven). The present paper studies the rate of escape of $V(n)$. We show that almost surely, the "lim\,inf'' expression in question is 0 if $\gamma\leq 1$, and is infinity otherwise. The corresponding problem for Brownian motion is also studied.

We present a new method to prove existence and uniform a priori estimates for Euclidean Gibbs measures corresponding to quantum anharmonic crystals. It is based first on the alternative characterization of Gibbs measures in terms of their logarithmic derivatives through integration by parts formulas, and second on the choice of appropriate Lyapunov functionals.

We consider the solution $(u,\eta)$ of the white-noise driven stochastic partial differential equation with reflection on the space interval $[0,1]$ introduced by Nualart and Pardoux, where $\eta$ is a reflecting measure on $[0,\infty)\times(0,1)$ which forces the continuous function u, defined on $[0,\infty)\times[0,1]$, to remain nonnegative and $\eta$ has support in the set of zeros of u. First, we prove that at any fixed time $t>0$, the measure $\eta([0,t]\times d\theta)$ is absolutely continuous w.r.t. the Lebesgue measure $d\theta$ on $(0,1)$. We characterize the density as a family of additive functionals of u, and we interpret it as a renormalized local time at $0$ of $(u(t,\theta))_{t\geq 0}$. Finally, we study the behavior of $\eta$ at the boundary of $[0,1]$. The main technical novelty is a projection principle from the Dirichlet space of a Gaussian process, vector-valued solution of a linear SPDE, to the Dirichlet space of the process u.

Let $X=(X_t)_{t \in S}$ be a real-valued Gaussian random process indexed by S with mean zero. General upper and lower estimates are given for the lower tail probability $\mathbb{P}(\sup_{t \in S} (X_t-X_{t_0}) \leq x )$ as $x \to 0$, with $t_0\in S$ fixed. In particular, sharp rates are given for fractional Brownian sheet. Furthermore, connections between lower tail probabilities for Gaussian processes with stationary increments and level crossing probabilities for stationary Gaussian processes are studied. Our methods also provide useful information on a random pursuit problem for fractional Brownian particles.

We prove an upper bound for the error in the exponential approximation of the hitting time law of a rare event in $\alpha$-mixing processes with exponential decay, $\phi$-mixing processes with a summable function $\phi$ and for general $\psi$-mixing processes with a finite alphabet. In the first case the bound is uniform as a function of the measure of the event. In the last two cases the bound depends also on the time scale t. This allows us to get further statistical properties as the ratio convergence of the expected hitting time and the expected return time. A uniform bound is a consequence. We present an example that shows that this bound is sharp. We also prove that second moments are not necessary for having the exponential law. Moreover, we prove a necessary condition for having the exponential limit law.

In this paper, we develop a strong Milstein approximation scheme for solving stochastic delay differential equations (SDDEs). The scheme has convergence order 1. In order to establish the scheme, we prove an infinite-dimensional Itô formula for "tame'' functions acting on the segment process of the solution of an SDDE. It is interesting to note that the presence of the memory in the SDDE requires the use of the Malliavin calculus and the anticipating stochastic analysis of Nualart and Pardoux. Given the nonanticipating nature of the SDDE, the use of anticipating calculus methods in the context of strong approximation schemes appears to be novel.

Let $X(t)$ be the symmetric $\alpha$-stable process in $\Bbb R^d (0<\alpha<2, d\ge 2)$. Then let $W(f)$ be the thorn $\{x \in \Bbb R^d\dvtx 0<x_1<1, (x_2^2 +\cdots + x_d^2)^{1/2} < f(x_1) \}$ where $f\dvtx (0,1)\rightarrow(0,1)$ is continuous, increasing with $f(0^+) = 0$. Recently Burdzy and Kulczycki gave an exact integral condition on f for the existence of a random time s such that $X(t)$ remains in the thorn $X(s)+ \overline{W(f)}$ for all $t \in [s,s+1)$. We extend their theorem to general open sets W with $0 \in \partial W$. In general, $\alpha$-processes may stay in sets which are quite lacunary and are not locally connected at 0.

The immortal branching Markov process (iBMP) is a modification of the usual branching model, in which each particle of generation n is counted, in addition to its offspring, as a member of generation $n+1$, its state being unchanged. When the number of offspring is Bernoulli, iBMP accounts, for instance, for the variability of the biological sequences that are produced by polymerase chain reactions (PCRs). This variability is due to the mutations and to the incomplete replications that affect the PCR. Estimators of PCR mutation rate and efficiency have been proposed that are based, in particular, on the mean empirical law $\eta_n$ of the mutations of a sequence. Unfortunately, $\eta_n$ is not analytically tractable. However, the infinite-population limit $\eta^*_n$ of $\eta_n$ is easily characterized in the two following, biologically relevant, cases. The Markovian kernel describes a homogeneous random walk, either on the integers or on some finite Cartesian product of a finite set. In the PCR context, this corresponds to infinite or finite targets, respectively. In this paper, we provide bounds of the discrepancy between $\eta_n$ and $\eta^*_n$ in these two cases. As a consequence, iBMP exhibits a strong averaging effect, even for surprisingly small starting populations. The bounds are explicit functions of the offspring law, the Markovian kernel, the number of steps n, the size of the initial population and, in the finite-target case, the size of the target. They concern every moment and, what might be less expected, the histogram itself. In the finite-target case, some of the bounds undergo a phase transition at an explicit value of the mutation rate per site and per cycle. We use precise estimates of the harmonic means of classical nondecreasing branching processes, whose proofs are included in the Appendix.

A notion of convergence in distribution for non (necessarily) measurable random elements, due to Hoffmann-Jørgensen, is characterized in terms of weak convergence of finitely additive probability measures. A similar characterization is given for a strengthened version of such a notion. Further, it is shown that the empirical process for an exchangeable sequence can fail to converge, due to the nonexistence of any measurable limit, although it converges for an i.i.d. sequence. Because of phenomena of this type, Hoffmann-Jørgensen's definition is extended to the case of a nonmeasurable limit. In the extended definition, naturally suggested by the main results, the limit is a finitely additive probability measure.

This paper obtains the weak convergence of the empirical processes of both the division points and the spacings that result from the Kakutani interval splitting model. In both cases, the limit processes are Gaussian. For the division points themselves, the empirical processes converge to a Brownian bridge as they do for the usual uniform splitting model, but with the striking difference that its standard deviations are about one-half as large. This result gives a clear measure of the degree of greater uniformity produced by the Kakutani model. The limit of the empirical process of the normalized spacings is more complex, but its covariance function is explicitly determined. The method of attack for both problems is to obtain first the analogous results for more tractable continuous parameter processes that are related through random time changes. A key tool in their analysis is an approximate Poissonian characterization that obtains for cumulants of a family of random variables that satisfy a specific functional equation central to the K-model.

By using probabilistic approaches, some uniform gradient estimates are obtained for Dirichlet heat semigroups on a Riemannian manifold with boundary. As an application, lower bound estimates of isoperimetric constants are presented in terms of functional inequalities.

An infinite-volume mixing or exponential-decay property in a spin system or percolation model reflects the inability of the influence of the configuration in one region to propagate to distant regions, but in some circumstances where such properties hold, propagation can nonetheless occur in finite volumes endowed with boundary conditions. We establish the absense [sic] of such propagation, particularly in two dimensions in finite volumes which are simply connected, under a variety of conditions, mainly for the Potts model and the Fortuin--Kasteleyn (FK) random cluster model, allowing external fields. For example, for the FK model in two dimensions we show that exponential decay of connectivity in infinite volume implies exponential decay in simply connected finite volumes, uniformly over all such volumes and all boundary conditions, and implies a strong mixing property for such volumes with certain types of boundary conditions. For the Potts model in two dimensions we show that exponential decay of correlations in infinite volume implies a strong mixing property in simply connected finite volumes, which includes exponential decay of correlations in simply connected finite volumes, uniformly over all such volumes and all boundary conditions.

Quantum stochastic calculus is extended in a new formulation in which its stochastic integrals achieve their natural and maximal domains. Operator adaptedness, conditional expectations and stochastic integrals are all defined simply in terms of the orthogonal projections of the time filtration of Fock space, together with sections of the adapted gradient operator. Free from exponential vector domains, our stochastic integrals may be satisfactorily composed yielding quantum Itô formulas for operator products as sums of stochastic integrals. The calculus has seen two reformulations since its discovery---one closely related to classical Itô calculus; the other to noncausal stochastic analysis and Malliavin calculus. Our theory extends both of these approaches and may be viewed as a synthesis of the two. The main application given here is existence and uniqueness for the Attal--Meyer equations for implicit definition of quantum stochastic integrals.

We show necessary and sufficient conditions for $R$-recurrence and \mbox{$R$-positivity} of one-dimensional diffusions killed at the origin. These conditions are stated in terms of the bottom eigenvalue function.

Let $B_n=(1/N)T_n^{1/2}X_nX_n^*T_n^{1/2}$ where $X_n=(X_{ij})$ is $n\times N$ with i.i.d. complex standardized entries having finite fourth moment, and $T_n^{1/2}$ is a Hermitian square root of the nonnegative definite Hermitian matrix $T_n$. The limiting behavior, as $n\to\infty$ with $n/N$ approaching a positive constant, of functionals of the eigenvalues of $B_n$, where each is given equal weight, is studied. Due to the limiting behavior of the empirical spectral distribution of $B_n$, it is known that these linear spectral statistics converges a.s. to a nonrandom quantity. This paper shows their rate of convergence to be $1/n$ by proving, after proper scaling, that they form a tight sequence. Moreover, if $\expp X^2_{11}=0$ and $\expp|X_{11}|^4=2$, or if $X_{11}$ and $T_n$ are real and $\expp X_{11}^4=3$, they are shown to have Gaussian limits.