We formulate and prove a local stable manifold theorem for stochastic differential equations (SDEs) that are driven by spatial Kunita-type semimartingales with stationary ergodic increments. Both Stratonovich and Itôtype equations are treated. Starting with the existence of a stochastic flow for a SDE, we introduce the notion of a hyperbolic stationary trajectory. We prove the existence of invariant random stable and unstable manifolds in the neighborhood of the hyperbolic stationary solution. For Stratonovich SDEs, the stable and unstable manifolds are dynamically characterized using forward and backward solutions of the anticipating SDE. The proof of the stable manifold theorem is based on Ruelle–Oseledec multiplicative ergodic theory.

By using the maximum principle and analysis of heat semigroups, Harnack inequalities are studied for log-Sobolev functions. From this, some lower bound estimates of the log-Sobolev constant are presented by using the spectral gap inequality and the coupling method. The resulting inequalities either recover or improve the corresponding ones proved by Chung and Yau. Especially, Harnack inequalities and estimates of log-Sobolev constants can be dimension-free.

The geometric stochastic analysis on the Riemannian path space developed recently gives rise to the concept of tangent processes. Roughly speaking, it is the infinitesimal version of the Girsanov theorem. Using this concept, we shall establish a formula of integration by parts on the path space over a loop group. Following the martingale method developed in Capitaine, Hsu and Ledoux, we shall prove that the logarithmic Sobolev inequality holds on the full paths. As a particular case of our result, we obtain the Driver–Lohrenz’s heat kernel logarithmic Sobolev inequalities over loop groups. The stochastic parallel transport introduced by Driver will play a crucial role.

In this paper, we construct a measure-valued diffusion on $D\subseteq \mathbb{R^d}$ whose underlying motion is a diffusion process with absorption at the boundary corresponding to an elliptic operator $$L = 1/2 \nabla \cdot a\nabla + b \cdot \nabla \text{ on } D \subseteq \mathbb{R}^d$$ and whose spatially dependent branching term is of the form $\beta(x)z-\alpha(x)z^2,x \in D$,where $\beta$ satisfies a very general condition and $\alpha> 0$. In the case that $\alpha$ and $\beta$ are bounded from above, we show that the measure-valued process can also be obtained as a limit of approximating branching particle systems.

We give criteria for extinction/survival, recurrence/transience of the support, compactness of the support, compactness of the range, and local extinction for the measure-valued diffusion. We also present a number of examples which reveal that the behavior of the measure-valued diffusion may be dramatically different from that of the approximating particle systems.

We give a characterization of $G$-regularity for super-Brownian motion and the Brownian snake. More precisely, we define a capacity on $E = (0,\infty) \times\mathbb{R}^d$, which is not invariant by translation. We then prove that the measure of hitting a Borel set $A \subset E$ for the graph of the Brownian snake excursion starting at (0,0) is comparable, up to multiplicative constants, to its capacity. This implies that super-Brownian motion started at time 0 at the Dirac mass $\varrho_0$ hits immediately $A$ [i.e., (0,0) is $G$ -regular for $A_c$] if and only if its capacity is infinite. As a direct consequence, if $Q \subset E$ is a domain such that (0,0) $\in\varrho Q$, we give a necessary and sufficient condition for the existence on $Q$ of a positive solution of $\varrho_tu + 1/2\Delta u = 2u^2$, which blows up at (0,0). We also give an estimate of the hitting probabilities for the support of super-Brownian motion at fixed time. We prove that if $d\geq 2$, the support of super-Brownian motion is intersection-equivalent to the range of Brownian motion.

We prove an almost sure invariance principle for diffusion driven by velocities with unbounded stationary vector potentials. The result generalizes to multiple particles motion, driven by a common velocity field and independent molecular Brownian motions.

In this paper we establish the existence and uniqueness of an $L^2(\mathbb{R})$ -valued solution for a one-dimensional Burgers’ equation perturbed by a space–time white noise on the real line. We show that the solution is continuous in space and time, provided the initial condition is continuous. The main ingredients of the proof are maximal inequalities for the stochastic convolution, and some a priori estimates for a class of deterministic parabolic equations.

We prove the existence and uniqueness, for any time, of a real-valued process solving a nonlinear stochastic wave equation driven by a Gaussian noise white in time and correlated in the two-dimensional space variable. We prove that the solution is regular in the sense of the Malliavin calculus. We also give a decay condition on the covariance function of the noise under which the solution has Hölder continuous trajectories and show that, under an additional ellipticity assumption, the law of the solution at any strictly positive time has a smooth density.

We continue the investigation of the behavior of the contact process on infinite connected graphs of bounded degree. Some questions left open by Salzano and Schonmann (1997) concerning the notions of complete convergence, partial convergence and the criterion $r = s$ are answered.

The continuity properties of the survival probability and the recurrence probability are studied. These order parameters are found to have a richer behavior than expected, with the possibility of the survival probability being discontinuous at or above the threshold for survival. A condition which guarantees the continuity of the survival probability above the survival point is introduced and exploited. The recurrence probability is shown to always be left-continuous above the recurrence point, and a necessary and sufficient condition for its right-continuity is introduced and exploited. It is shown that for homogeneous graphs the survival probability can only be discontinuous at the survival point, and the recurrence probability can only be discontinuous at the recurrence point.

For graphs which are obtained by joining a finite number of severed homogeneous trees by means of a finite number of vertices and edges, the survival point, the recurrence point and the discontinuity points of the survival and recurrence probabilities are located.

We study several statistical mechanical models on a general tree. Particular attention is devoted to the classical Heisenberg models, where the state space is the $d$-dimensional unit sphere and the interactions are proportional to the cosines of the angles between neighboring spins. The phenomenon of interest here is the classification of phase transition (non-uniqueness of the Gibbs state) according to whether it is robust. In many cases, including all of the Heisenberg and Potts models, occurrence of robust phase transition is determined by the geometry (branching number) of the tree in a way that parallels the situation with independent percolation and usual phase transition for the Ising model. The critical values for robust phase transition for the Heisenberg and Potts models are also calculated exactly. In some cases, such as the $q \geq 3$ Potts model, robust phase transition and usual phase transition do not coincide, while in other cases, such as the Heisenberg models, we conjecture that robust phase transition and usual phase transition are equivalent. In addition, we show that symmetry breaking is equivalent to the existence of a phase transition, a fact believed but not known for the rotor model on $\mathbb{Z}^2$ .

We study the eigenvalue distribution of $N \times N$ symmetric random matrices $H_N (x, y) = N^-1/2h(x,y),x,y=1,\ldots,N$, where $h(x, y), x \leq y$ are Gaussian weakly dependent random variables. We prove that the normalized eigenvalue counting function of $H_{N}$ converges with probability 1 to a nonrandom function $\mu(\lambda)$ as $N\rightarrow\infty$. We derive an equation for the Stieltjes transform of the measure $d\mu(\lambda)$ and show that the latter has a compact support $\Lambda_\mu$. We find the upper bound for $\lim\sup_{N\rightarrow\infty}\|H_N\|$ and study asymptotically the case when there are no eigenvalues of $H_N$ outside of $\Lambda_\mu$ when $N \rightarrow \infty$.

For a simple point process $\Xi$ on a suitable topological space, the associated Palm distribution at a point s may be approximated by the conditional distribution, given that $\Xi$ hits a small neighborhood of $s$. To study the corresponding approximation problem for more general random sets, we develop a general duality theory, which allows the Palm distributions with respect to an associated random measure to be expressed in terms of conditional densities with suitable martingale and continuity properties. The stated approximation property then becomes equivalent to a certain asymptotic relation involving conditional hitting probabilities. As an application, we consider the Palm distributions of regenerative sets with respect to their local time random measures.

Let ${X _n}$ be a transient reversible Markov chain and let $f$ be a function on the state space with finite Dirichlet energy. We prove crossing inequalities for the process ${f (X _n)}_{n\geq 1}$ and show that it converges almost surely and in $L^2$. Analogous results are also established for reversible diffusions on Riemannian manifolds.

Two classes of processes are considered. One is a class of spectrally positive infinitely divisible processes which includes all such stable processes. The other is a class of processes constructed from the sequence of partial sums of independent identically distributed positive random variables. A condition analogous to regular variation of the tails is imposed. Then a large deviation principle and a Strassen-type law of the iterated logarithm are presented. These theorems focus on unusually large values of the processes. They are expressed in terms of Skorokhod’s $M_1$ topology.

If $X$ is integrable, $F$ is its cdf and $F_n$ is the empirical cdf based on an i.i.d. sample from $F$, then the Wasserstein distance between $F_n$ and $F$, which coincides with the $L_1$ norm $\int_-\infty^\infty|F_n(t)|dt$ of the centered empirical process, tends to zero a.s. The object of this article is to obtain rates of convergence and distributional limit theorems for this law of large numbers or, equivalently, stochastic boundedness and distributional limit theorems for the $L_1$ norm of the empirical process. Some limit theorems for the Ornstein–Uhlenbeck process are also derived as a by-product.

We consider quadratic forms $$Q_n = \sum_{1 \le j \neq k \le n} a_{jk}X_j X_k,$$ where $X_j$ are i.i.d. random variables with finite third moment. We obtain optimal bounds for the Kolmogorov distance between the distribution of $Q_n$ and the distribution of the same quadratic forms with $X_j$ replaced by corresponding Gaussian random variables. These bounds are applied to Toeplitz and random matrices as well as to nonstationary AR(1) processes.