Cox and Durrett proved the shape theorem of epidemics and forest fires with nearest neighbor interactions. They also conjectured that the shape theorem is still true with finite range interactions. In this paper, we answer this conjecture affirmatively. The method we develop allows us also to improve the result of Zhang and Zhang for all dimension.

We prove conservation of local equilibrium for attractive particle systems. Our method applies as well to gradient asymmetric processes with mean drift 0 under diffusive $(N^2)$ rescaling. The hydrodynamical behavior is proved for bounded continuous initial profiles under Euler $(N)$ rescaling and for bounded a.s. continuous profiles under diffusive rescaling. We prove that, for attractive systems, the conservation of local equilibrium follows from a law of large numbers for the density field.

In this paper, we apply the Brydges-Spencer lace expansion and the Hara-Slade analysis to obtain the triangle condition for the nearest-neighbor oriented bond percolation in high dimensions and for the spread-out oriented bond percolation in $Z^d \times Z, d \geq 5$. Furthermore, we also establish the infrared bound in the subcritical region and the mean-field behavior for these models.

We present a new approach to the principle of large deviations for the empirical field of a Gibbsian random field on the integer lattice $\mathbb{Z}^d$. This approach has two main features. First, we can replace the traditional weak topology by the finer topology of convergence of cylinder probabilities, and thus obtain estimates which are finer and more widely applicable. Second, we obtain as an immediate consequence a limit theorem for conditional distributions under conditions on the empirical field, the limits being those predicted by the maximum entropy principle. This result implies a general version of the equivalence of Gibbs ensembles, stating that every microcanonical limiting state is a grand canonical equilibrium state. We also prove a converse to the last statement, and discuss some applications.

We investigate large deviations for the empirical distribution functional of a Gaussian random field on $\mathbb{R}^{\mathbb{Z}^d}, d \geq 3$, in the phase transition regime. We first prove that the specific entropy governs an $N^d$ volume order large deviation principle outside the Gibbsian class. Within the Gibbsian class we derive an $N^{d-2}$ capacity order large deviation principle with exact rate function, and we apply this result to the asymptotics of microcanonical ensembles. We also give a spins' profile description of the field and show that smooth profiles obey $N^{d-2}$ order large deviations, whereas discontinuous profiles obey $N^{d-1}$ surface order large deviations.

For strictly stationary random fields indexed by $\mathbb{R}^d$ or $\mathbb{Z}^d$, certain versions of the "strong mixing" condition are equivalent to corresponding versions of the "$\rho$-mixing" condition.

Limiting distributions of square-integrable infinite order U-statistics were first studied by Dynkin and Mandelbaum and Mandelbaum and Taqqu. We extend their results to the case of non-Poisson random sample size. Multiple integrals of non-Gaussian generalized fields are constructed to identify the limiting distributions. An invariance principle is also established. We use these results to study the limiting distribution of the amount of charge left in some set by an infinite system of signed Markovian particles when the initial particle density goes to infinity. By selecting the initial particle distribution, we determine the limiting distribution of charge, constructing different non-Gaussian generalized random fields, including Laplace, $\alpha$-stable and their multiple integrals.

We consider two critical spatial branching processes on $\mathbb{R}^d$: critical branching Brownian motion, and the Dawson-Watanabe process. A basic feature of these processes is that their ergodic behavior is highly dimension-dependent. It is known that in low dimensions, $d \leq 2$, the unique invariant measure with finite intensity is $\delta_0$, the unit point mass on the empty state. In high dimensions, $d \geq 3$, there is a one-parameter family of nondegenerate invariant measures. We prove here that for $d \leq 2, \delta_0$ is the only invariant measure. In our proof we make use of sub- and super-solutions of the partial differential equation $\partial u/\partial t = (1/2) \Delta u - bu^2$.

Based on the well-known Borell inequality and on a general theorem for large and small increments of Banach space valued stochastic processes of Csaki, Csorgo and Shao, we establish some almost sure path behaviour of increments in general, and moduli of continuity in particular, for $l^p$-valued, $1 \leq p < \infty$, Gaussian processes with stationary increments. Applications to $l^p$-valued fractional Wiener and Ornstein-Uhlenbeck processes are also discussed. Our results refine and extend those of Csaki, Csorgo and Shao.

In this paper we establish some general forms of the law of the iterated logarithm for independent random variables $(X_n)$ with Banach space values, where $(X_n)$ is not necessarily identically distributed. Our results include the Kolmogorov law of the iterated logarithm (LIL) in both finite and infinite dimensional cases, and they improve the Wittmann LIL as well as extend it to the vector setting. The Ledoux-Talagrand LIL for an i.i.d. sequence is also a simple corollary of our results.

A general bounded law of the iterated logarithm for Banach space valued random variables is established. Our results implies: (a) the bounded LIL of Ledoux and Talagrand, (b) a bounded LIL for random variables in the domain of attraction of a Gaussian law and (c) new LIL results for random variables outside the domain of attraction of a Gaussian law in cases where the classical norming sequence $\{\sqrt{nLLn}\}$ does not work. Basic ingredients of our proof are an infinite-dimensional Fuk-Nagaev type inequality and an infinite-dimensional version of Klass's $K$-function.

Let $X = \sum\varepsilon_nx_n$ be a Rademacher series with vector-valued coefficients. We obtain an approximate formula for the distribution of the random variable $\|X \|$ in terms of its mean and a certain quantity derived from the $K$-functional of interpolation theory. Several applications of the formula are given.

We consider an exchangeably weighted bootstrap of the general function-indexed empirical process. We find sufficient conditions on the bootstrap weights for the central limit theorem to hold for the bootstrapped empirical process, almost surely and in probability. The results resemble those of Gine and Zinn for Efron's bootstrap. As a corollary we obtain a result on the almost sure convergence in distribution of the Efron-bootstrapped empirical process with arbitrary sample size. A large number of bootstrap resampling schemes emerge as special cases of our results.

Local limit theorems and saddlepoint approximations are given for random walks on a free group whose step distributions have finite support. The techniques used to prove these results are necessarily different from those used for random walks in Euclidean spaces, because Fourier analysis is not available; the basic tools are the elementary theory of algebraic functions and the Perron-Frobenius theory of nonnegative matrices. An application to the structure of the boundary process is also given.

We develop techniques for bounding the rate of convergence of a symmetric random walk on a finite group to the uniform distribution. The techniques gives bounds on the second largest (and other) eigenvalues in terms of the eigenvalues of a comparison chain with known eigenvalues. The techniques yield sharp rates for a host of previously intractable problems on the symmetric group.

A stationary process $\{X_n\}_{n \in \mathbb{Z}}$ is said to be $k$-dependent if $\{X_n\}_{n < 0}$ is independent of $\{X_n\}_{n > k-1}$. It is said to be a $k$-block factor of a process $\{Y_n\}$ if it can be represented as $X_n = f(Y_n,\ldots, Y_{n+k-1}),$ where $f$ is a measurable function of $k$ variables. Any $(k + 1)$-block factor of an i.i.d. process is $k$-dependent. We answer an old question by showing that there exists a one-dependent process which is not a $k$-block factor of any i.i.d. process for any $k$. Our method also leads to generalizations of this result and to a simple construction of an eight-state one-dependent Markov chain which is not a two-block factor of an i.i.d. process.

We study shift transformations on a general abstract Wiener space $(E, H, \mu)$, which have the form: $E \ni \omega \mapsto \mathscr{J}^\phi\omega \equiv \omega - \int^T_0 \phi_t(\omega)Z(dt) \in E,$ where $\phi_t(\omega)$ is a scalar function on $\lbrack 0, T\rbrack \times E$ and $Z$ is an orthogonal $H$-valued measure. Under suitable conditions for the kernel $\phi$, we construct explicitly a probability measure $\mu^\phi$ on $E$, which is equivalent to the standard Wiener measure $\mu$ and has the property: $\mu^\phi\{\mathscr{F}^\phi \in A\} = \mu(A), A \in \mathscr{B}_E$. The main result presents an analog of the well-known Cameron-Martin-Girsanov theorem for the case where the shift is allowed to anticipate. This leads to an additional integral term in the Girsanov exponent. Also, the Wiener-Ito integral in this exponent is now replaced by an extended stochastic integral.

We consider the heat equation with a noise term, on a finite interval with periodic boundary conditions. We show how to construct coupled solutions to the equation. As applications, we prove the uniqueness of invariant measures and the triviality of bounded harmonic functions.

Let $Y$ be a random variable defined by a polynomial $p(W)$ of degree $n$ in finitely many normally distributed variables. This paper studies which such variables $Y$ are "determinate," i.e., have probability laws uniquely determined by their moments. Extending results of Berg, which applied to powers of a single normal variable, we prove that (a) $Y$ is determinate if $n = 1, 2$ or if $n = 4$, with the essential support of the law of $Y$ strictly smaller than the real line, and (b) $Y$ is not determinate either if $n$ is odd $\geq 3$ or if $n$ is even $\geq 6$ such that $p(\mathbf{w})$ attains a finite minimum value. Some other polynomials $Y = p(\mathbf{W})$ with even degree $n \geq 4$ are proved not to be determinate.

A number of asymptotic problems for "classical" stochastic processes leads to diffusion processes on graphs. In this paper we study several such examples and develop a general technique for these problems. Diffusion in narrow tubes, processes with fast transmutations and small random perturbations of Hamiltonian systems are studied.

We extend Echeverria's criterion for invariant measures for a Markov process characterized via martingale problems to the case where the state space of the Markov process is a complete separable metric space. Essentially, the only additional conditions required are a separability condition on the operator occurring in the martingale problem and the well-posedness of the martingale problem in the class of progressively measurable solutions (as opposed to well-posedness in the class of r.c.l.l. solutions, i.e. solutions with paths that are right continuous and have left limits, in the locally compact case). Uniqueness of the solution to the (measure valued) evolution equation for the distribution of the Markov process (as well as a perturbed equation) is also proved when the test functions are taken from the domain of the operator of the martingale problem.