This paper is concerned with the integration (of 1-forms) against the Markov stochastic process associated with a second-order elliptic differential operator in divergence form. It focuses on the limiting behavior of the integral as the process leaves a fixed point or goes to infinity. This extends previous work in the area where advantage was usually taken of the fact that the operator was self adjoint and started with the associated invariant measure. Applications are given. For example, it is a trivial consequence that the diffusion associated to a uniformly elliptic operator on a negatively curved Cartan–Hadamard manifold has an asymptotic direction (recovering and strengthening the previous arguments of Pratt, Sullivan and others). The approach can also be used to construct a Lévy area for such processes, to study the thinness of sets for the elliptic operator, and probably has wider applications.

We study an ordinary differential equation controlled by a stochastic process. We present results on existence and uniqueness of solutions, on associated local times (Trotter and Ray–Knight theorems) and on time and direction of bifurcation. A relationship with Lipschitz approximations to Brownian paths is also discussed.

We study asymptotic growth rates of stochastic flows on $\mathbf{R}^d$ and their derivatives with respect to the spatial parameter under Lipschitz conditions on the local characteristics of the generating semimartingales. In a first step these conditions are seen to imply moment inequalities for the flow $\phi$ of the form $$E \sup_{0 \le t \le T}|\phi_{0t} (x) - \phi_{0t} (y)|^p\le|x-y|^p\exp(cp^2)\quad\text{for all $p\ge 1$.}$$ In a second step we deduce the growth rates from an integrated version of these moment inequalities, using the continuity lemma of Garsia, Rodemich and Rumsey. We provide two examples to show that our results are sharp.

Let $dx=\sum_{i=0}^{m}A_ix\circ dW^i$ be a linear SDE in $\mathbb{R}^d$, generating the flow $\Phi_t$ of linear isomorphisms. The multiplicative ergodic theorem asserts that every vector $v\in\mathbb{R}^d\backslash\{0\}$ possesses a Lyapunov exponent (exponential growth rate) $\lambda(v)$ under $\Phi_t$, which is a random variable taking its values from a finite list of canonical exponents $\lambda_i$ realized in the invariant Oseledets spaces $E_i$. We prove that, in the case of simple Lyapunov spectrum, every 2-plane $p$ in $\mathbb{R}^d$ possesses a rotation number $\rho(p)$ under $\Phi_t$ which is defined as the linear growth rate of the cumulative inffinitesimal rotations of a vector $v_t$ inside $\Phi_t(p)$. Again, $\rho(p)$ is a random variable taking its values from a finite list of canonical rotation numbers $\rho_{ij}$ realized in span $(E_i, E_j)$. We give rather explicit Furstenberg-Khasminski-type formulas for the $\rho_{i,j}$. This carries over results of Arnold and San Martin from random to stochastic differential equations, which is made possible by utilizing anticipative calculus.

Let $L$ be a local time. It is well known that there exist a law of the iterated logarithm and a modulus of continuity for $L$. Motivated by the case of real Brownian motion, we study the existence of fast points and slow points of $L$. We prove the existence of such points by considering the right-continuous inverse of $L$, which is a subordinator.

Models of populations in which a type or location, represented by a point in a metric space $E$, is associated with each individual in the population are considered. A population process is neutral if the chances of an individual replicating or dying do not depend on its type. Measure-valued processes are obtained as infinite population limits for a large class of neutral population models, and it is shown that these measure-valued processes can be represented in terms of the total mass of the population and the de Finetti measures associated with an $E^{\infty}$ -valued particle model $X=(X_1, X_2\ldots)$ such that, for each $t \geq 0,(X_1(t),X_2(t),\ldots)$ is exchangeable. The construction gives an explicit connection between genealogical and diffusion models in population genetics. The class of measure-valued models covered includes both neutral Fleming-Viot and Dawson-Watanabe processes. The particle model gives a simple representation of the Dawson-Perkins historical process and Perkins’s historical stochastic integral can be obtained in terms of classical semimartingale integration. A number of applications to new and known results on conditioning, uniqueness and limiting behavior are described.

It is known that the contact process on a homogeneous tree of degree $d+1\geq3$ has a weak survival phase, in which the infection survives with positive probability but nevertheless eventually vacates every finite subset of the tree. It is shown in this paper that in the weak survival phase there exists a spherically symmetric invariant measure whose density decays exponentially at infinity, thus confirming a conjecture of Liggett. The proof is based on a study of the relationships between various thermodynamic parameters and functions associated with the contact process initiated by a single infected site. These include (1) the growth profile, which determines the exponential rate of growth in space-time on the event of survival, (2) the exponential rate $\beta$ of decay of the hitting probability function at infinity (also studied by the author) and (3) the exponential rate $\beta$ of decay in time $t$ of the probability that the initial infected site is infected at time $t$. It is shown that $\beta$ is a strictly increasing function of the infection rate $\lambda$ in the weak survival phase, and that $\beta = 1/\sqrt{d}$ at the upper critical point $\lambda_2$ demarcating the boundary between the weak 2 and strong survival phases. It is also shown that $\eta < 1$ except at $\lambda_2$, where $\eta =1$.

We consider the $d$-dimensional threshold voter model. It is known that, except in the one-dimensional nearest-neighbor case, coexistence occurs (nontrivial invariant measures exist). In fact, there is a nontrivial limit $\eta_\infty^{1/2}$ obtained by starting from the product measure with density 1/2. We show that in these coexistent cases, $$\eta_t \Rightarrow \alpha\delta_0 + \beta\delta_1 + (1 - \alpha - \beta)\eta^{1/2}_\infty \quad \text{as } t \to \infty,$$ where $\alpha=P(\tau_0<\infty), \beta=P(\tau_1<\infty), \tau_0$ and $\tau_1$ are the first hitting times of the all-zero and all-one configurations, respectively, and $\Rightarrow$ denotes weak convergence.

Suppose $X_1, X_2, X_3,\ldots$ are independent random points in $\mathbf{R}^d,d\geq 2$, with common density $f$, having connected compact support $\Omega$ with smooth boundary $\partial\Omega$, with $f|_{\Omega}$ continuous. Let $M_{n}$ denote the smallest $r$ such that the union of balls of diameter $r$ centered at the first $n$ points is connected. Let $\theta$ denote the volume of the unit ball. Then as $n\to\infty$, $$n\theta M^d_n/\log n \to \max\Big(\big(\min\limits_{\Omega}f\big)^{-1},2(1 - 1/d)\big(\min\limits_{\partial\Omega}f\big)^{-1}\Big),\quad\text{a.s.}$$

Let $B$ be a standard one-dimensional Brownian motion started at 0. Let $L_{t,v}(|B|)$ be the occupation density of $|B|$ at level $v$ up to time $t$. The distribution of the process of local times $(L_{t,v}(|B|)v\geq0)$ conditionally given $B_{t}= 0$ and $L_{t,0}(|B|)=l$ is shown to be that of the unique strong solution $X$ of the Itô SDE, $$dX_v = \Big\{4 - X^2_v\Big(t - \int_0^v X_u\,du\Big)^{-1}\Big\}dv + 2\sqrt{X_v}\,dB_v$$ on the interval $[0,V_{t}(X))$, where $V_{t}(X):= \inf\{v:\int_0^v X_u,du=t\}$, and $X_v= 0$ for all $v\geq V_t(X)$. This conditioned from study of the Ray-Knight description of Brownian local times arises from study of the asymptotic distribution as $n\rightarrow\infty$ and $2k/\sqrt{n}\rightarrow l$ of the height profile of a uniform rooted random forest of $k$ trees labeled by a set of $n$ elements, as obtained by conditioning a uniform random mapping of the set to itself to have $k$ cyclic points. The SDE is the continuous analog of a simple description of a Galton-Watson branching process conditioned on its total progeny. For $l = 0$, corresponding to asymptotics of a uniform random tree, the SDE gives a description of the process of local times of a Brownian excursion which is equivalent to Jeulin’s description of these local times as a time change of twice a Brownian excursion. Another corollary is the Biane-Yor description of the local times of a reflecting Brownian ridge as a time-changed reversal of twice a Brownian meander of the same length.

We establish the large deviation principle for stochastic differential equations with averaging in the case when all coefficients of the fast component depend on the slow one, including diffusion.

The main theorem is the large deviation principle for the doubly indexed sequence of random measures $$W_{r,q}(dx \times dy) \doteq \theta(dx) \otimes \textstyle\sum\limits_{k=1}^{2^r} 1_{D_{r,k}}(x) L_{q,k} (dy).$$ Here $\theta$ is a probability measure on a Polish space $\mathscr{X},\{D_{r,k}k=1,\ldots,2^r\}$ is a dyadic partition of $\mathscr{X}$ (hence the use of $2^r$ summands) satisfying $\theta\{D_{r,k}\}= 1/2^r$ and $L_{q,1}L_{q,2},\ldots , L_{q,2_r}$ is an independent, identically distributed sequesnce of random probability measures on a Ploish space $ \mathscr{Y}$ such that $\{L_{q,k}q\in \mathbb{N}\}$ satisfies the large deviation principle with a convex rate function. A number of related asymptotic results are also derived.

The random measures $W_{ r,q}$ have important applications to the statistical mechanics of turbulence. In a companion paper, the large deviation principle presented here is used to give a rigorous derivation of maximum entropy principles arising in the well-known Miller–Robert theory of two-dimensional turbulence as well as in a modification of that theory recently proposed by Turkington.

Under mild assumptions we prove that for any local function $u$ the decay rate to equilibrium in the variance sense of zero range dynamics on $d$-dimensional integer lattice is $C_u t^{-d/2}+ o(t^{-d/2})$. The constant $C_u$ is computed explicitly.

In a totally asymmetric simple $K$-exclusion process, particles take nearest-neighbor steps to the right on the lattice Z, under the constraint that each site contain at most $K$ particles. We prove that such processes satisfy hydrodynamic limits under Euler scaling, and that the limit of the empirical particle profile is the entropy solution of a scalar conservation law with a concave flux function. Our technique requires no knowledge of the invariant measures of the process, which is essential because the equilibria of asymmetric $K$-exclusion are unknown. But we cannot calculate the flux function precisely. The proof proceeds via a coupling with a growth model on the two-dimensional lattice. In addition to the basic $K$-exclusion with constant exponential jump rates, we treat the sitedisordered case where each site has its own jump rate, randomly chosen but frozen for all time. The hydrodynamic limit under site disorder is new even for the simple exclusion process (the case $K = 1$). Our proof makes no use of the Markov property, so at the end of the paper we indicate how to treat the case with arbitrary waiting times.

We study the question on how large the size of singular component of translates of product probability measures can be in terms of Fisher information. We then prove some shift inequalities.

This paper concerns moment and tail probability inequalities and the strong law of large numbers for $U$-statistics with nonnegative or symmetrized kernels and their multisample and decoupled versions. Sub-Bernoulli functions are used to obtain the moment and tail probability inequalities, which are then used to obtain necessary and sufficient conditions for the almost sure convergence to zero of normalized $U$-statistics with nonnegative or completely symmetrized kernels, without further regularity conditions on the kernel or the distribution of the population, for normalizing constants satisfying a simple condition. Moments of $U$-statistics are bounded from above and below by that of maxima of certain kernels, up to scaling constants. The multisample and decoupled versions of these results are also considered.

Let $X,X_1,X_2,\ldots$ be i.i.d. random variables taking values in a measurable space $\mathscr{X}$. Let $\phi(x,y)$ and $\phi_1(x)$ denote measurable functions of the arguments $x,y\in\mathscr{X}$. Assuming that the kernel $\phi$ is symmetric and the $\mathbf{E}\phi(x,X)= 0$, for all $x$, and $\mathbf{E}\phi_1(X) = 0$, we consider $U$-statistics of type $$T = N^{-1} \textstyle\sum\limits_{1\le j < k\le N} \phi(X_j,X_k) + N^{-1/2} \textstyle\sum\limits_{1\le j\le N}\phi_1(X_j). $$ Is is known that the conditions $\mathbf{E}\phi^2(X,X_1)<\infty$ and $\mathbf{E}\phi_1^2(X)<\infty$ imply that the distribution function of $T$, say $F$, has a limit, say $F_0$, which can be described in terms of the eigenvalues of the Hilbert-Schmidt operator associated with the kernel $\phi(x,y)$. Under optimal moment conditions, we prove that $$\Delta_N = \sup_x |F(x) - F_0(x) - F_1(x)|= \mathcal{O}(N^{-1}), $$ provided that at least nine eigenvalues of the operator do not vanish. Here $F_1$ denotes an Edgeworth-type correction. We provide explicit bounds for $\Delta_N$ and for the concentration functions of statistics of type $T$.

It is a well-known fact that the periodogram ordinates of an iid mean-zero Gaussian sequence at the Fourier frequencies constitute an iid exponential vector, hence the maximum of these periodogram ordinates has a limiting Gumbel distribution. We show for a non-Gaussian iid mean-zero, finite variance sequence that this statement remains valid. We also prove that the point process constructed from the periodogram ordi-nates converges to a Poisson process. This implies the joint weak convergence of the upper order statistics of the periodogram ordinates. These results are in agreement with the empirically observed phenomenon that various functionals of the periodogram ordinates of an iid finite variance sequence have very much the same asymptotic behavior as the same functionals applied to an iid exponential sample.

In this paper we introduce a technique for obtaining exponential inequalities, with particular emphasis placed on results involving ratios. Our main applications consist of approximations to the tail probability of the ratio of a martingale over its conditional variance (or its quadratic variation for continuous martingales). We provide examples that strictly extend several of the classical exponential inequalities for sums of independent random variables and martingales. The spirit of this application is that, when going from results for sums of independent random variables to martingales, one should replace the variance by the conditional variance and the exponential of a function of the variance by the expectation of the exponential of the same function of the conditional variance. The decoupling inequalities used to attain our goal are of independent interest. They include a new exponential decoupling inequality with constraints and a sharp inequality for the probability of the intersection of a fixed number of dependent sets. Finally, we also present an exponential inequality that does not require any integrability conditions involving the ratio of the sum of conditionally symmetric variables to its sum of squares.

Let $\eta$ be a stationary Harris recurrent Markov chain on a Polish state space $(S, \mathscr{F})$, with stationary distribution $\mu$. Let $\Psi_n:=\sum_{i-1}^n I\{\mu\in S_1\}$ be the number of visits to $S_1\in\mathscr{F}$ by $\eta$, where $S_1$ is “rare” in the sense that $\mu(S_1)$ is “small.” We want to find an approximating compound Poisson distribution for$ \mathscr{L}(\Psi_n)$, such that the approximation error, measured using the total variation distance, can be explicitly bounded with a bound of order not much larger than $\mu(S_1)$. This is motivated by the observation that approximating Poisson distributions often give larger approximation errors when the visits to $S_1$ by $\eta$ tend to occur in clumps and also by the compound Poisson limit theorems of classical extreme value theory.

We here propose an approximating compound Poisson distribution which in a natural way takes into account the regenerative properties of Harris recurrent Markov chains. A total variation distance error bound for this approximation is derived, using the compound Poisson Stein equation of Barbour, Chen and Loh and certain couplings. When the chain has an atom $S_0$ (e.g., a singleton) such that $\mu(S_0)>0$, the bound depends only on much studied quantities like hitting probabilities and expected hitting times, which satisfy Poisson’s equation. As “by-products” we also get upper and lower bounds for the error in the approximation with Poisson or normal distributions. The above results are illustrated by numerical evaluations of the error bound for some Markov chains on finite state spaces.