We study a class of (nonsymmetric) Dirichlet forms $(\mathscr{E}, D(\mathscr{E}))$ having a space of measures as state space $E$ and derive some general results about them. We show that under certain conditions they "generate" diffusion processes $\mathbf{M}$. In particular, if $\mathbf{M}$ is ergodic and $(\mathscr{E}, D(\mathscr{E}))$ is symmetric w.r.t. quasi-every starting point, the large deviations of the empirical distribution of $\mathbf{M}$ are governed by $\mathscr{E}$. We apply all of this to construct Fleming-Viot processes with interactive selection and prove some results on their behavior. Among other things, we show some support properties for these processes using capacitary methods.

In a one-dimensional single point-catalytic continuous super-Brownian motion studied by Dawson and Fleischmann, the occupation density measure $\lambda^c$ at the catalyst's position $\mathcal{C}$ is shown to be a singular (diffuse) random measure. The source of this qualitative new effect is the irregularity of the varying medium $\delta_\mathcal{C}$ describing the point catalyst. The proof is based on a probabilistic characterization of the law of the Palm canonical clusters $\chi$ appearing in the Levy-Khintchine representation of $\lambda^\mathcal{C}$ in a historical process setting and the fact that these $\chi$ have infinite left upper density (with respect to Lebesgue measure) at the Palm time point.

A slow point from the left for Brownian motion is a time during a given interval for which the oscillations of the path immediately to the left of this time are smaller than the typical ones, that is, those given by the local LIL. These slow points occur at random times during a given interval. For historical super-Brownian motion, the support at a fixed time contains an infinite collection of paths. This paper makes use of a branching process description of the support to investigate the slowness of these paths at the fixed time. The upper function found is the same as that found for slow points in the Brownian motion case.

Internal diffusion limited aggregation (internal DLA) is a cluster model in $\mathbb{Z}^d$ where new points are added by starting random walkers at the origin and letting them run until they have found a new point to add to the cluster. It has been shown that the limiting shape of internal DLA clusters is spherical. Here we show that for $d \geq 2$ the fluctuations are subdiffusive; in fact, that they are of order at most $n^{1/3}$, at least up to logarithmic corrections. More precisely, we show that for all sufficiently large $n$ the cluster after $m = \lbrack\omega_dn^d\rbrack$ steps covers all points in the ball of radius $n - n^{1/3}(\ln n)^2$ and is contained in the ball of radius $n + n^{1/3}(\ln n)^4$.

The structure of a spanning forest that generalizes the minimal spanning tree is considered for infinite graphs with a value $f(b)$ attached to each bond $b$. Of particular interest are stationary random graphs; examples include a lattice with iid uniform values $f(b)$ and the Voronoi or complete graph on the sites of a Poisson process, with $f(b)$ the length of $b$. The corresponding percolation models are Bernoulli bond percolation and the "lily pad" model of continuum percolation, respectively. It is shown that under a mild "simultaneous uniqueness" hypothesis, with at most one exception, each tree in the forest has one topological end, that is, has no doubly infinite paths. If there is a tree in the forest, necessarily unique, with two topological ends, it must contain all sites of an infinite cluster at the critical point in the corresponding percolation model. Trees with zero, or three or more, topological ends are not possible. Applications to invasion percolation are given. If all trees are one-ended, there is a unique optimal (locally minimax for $f$) path to infinity from each site.

We study the behavior of random walk in random environment (RWRE) on trees in the critical case left open in previous work. Representing the random walk by an electrical network, we assume that the ratios of resistances of neighboring edges of a tree $\Gamma$ are i.i.d. random variables whose logarithms have mean zero and finite variance. Then the resulting RWRE is transient if simple random walk on $\Gamma$ is transient, but not vice versa. We obtain general transience criteria for such walks, which are sharp for symmetric trees of polynomial growth. In order to prove these criteria, we establish results on boundary crossing by tree-indexed random walks. These results rely on comparison inequalities for percolation processes on trees and on some new estimates of boundary crossing probabilities for ordinary mean-zero finite variance random walks in one dimension, which are of independent interest.

Let $S$ be a countable set and $\Lambda$ the collection of all subsets of $S$. We consider interacting particle systems (IPS) $\{\eta_k\}$ on $\Lambda$, with duals $\{\tilde\eta_t\}$ and duality equation $P\lbrack |\eta^\zeta_t \cap A| \operatorname{odd} = \tilde{P}\lbrack |\tilde\eta_t^A \cap \zeta| \operatorname{odd} \rbrack, \zeta, A \subset S, A$ finite Under certain conditions we find all the extreme invariant distributions that arise as limits of translation invariant initial configurations. Specific systems will be considered. A new property of the annihilating particle model is then used to prove a limiting relation between the annihilating and coalescing particle models.

It is shown that the transition semigroup $(P_t)_{t\geq0}$ corresponding to a nonlinear stochastic evolution equation is strong Feller and irreducible, provided the nonlinearities are Lipschitz continuous and the diffusion term is nondegenerate. This result ensures the uniqueness of the invariant measure for $(P_t)_{t\geq0}$.

Let $\rho_t$ be the radial part of a Brownian motion in an $n$-dimensional Riemannian manifold $M$ starting at $x$ and let $T = T_\varepsilon$ be the first time $t$ when $\rho_t = \varepsilon$. We show that $E\lbrack \rho^2_{t\wedge T} \rbrack = nt - (1/6)S(x)t^2 + \sigma(t^2)$, as $t \downarrow 0$, where $S(x)$ is the scalar curvature. The same formula holds for $E\lbrack\rho^2_t\rbrack$ under some boundedness condition on $M$.

The solution $u(t, x)$ of a parabolic stochastic partial differential equation is a random element of the space $\mathscr{E}_{\alpha,\beta}$ of Holder continuous functions on $\lbrack 0, T \rbrack \times \lbrack 0, 1 \rbrack$ of order $\alpha = \frac{1}{4} - \varepsilon$ in the time variable and $\beta = \frac{1}{2} - \varepsilon$ in the space variable, for any $\varepsilon > 0$. We prove a support theorem in $\mathscr{E}_{\alpha,\beta}$ for the law of $u$. The proof is based on an approximation procedure in Holder norm (which should have its own interest) using a space-time polygonal interpolation for the Brownian sheet driving the SPDE, and a sequence of absolutely continuous transformations of the Wiener space.

We obtain a sharp probability bound on the maximal function of a strong subordinate of a bounded submartingale. An analogous inequality also holds for stochastic integrals in which the integrator is a bounded submartingale and the integrand is a bounded predictable process.

We are investigating Markov process expectations for large time of the form $\exp(TF(L_T))$, where $L_T$ is the empirical measure of a uniformly ergodic Markov process and $F$ is a smooth functional. Such expressions are evaluated to a factor which converges to 1. In contrast to earlier work on the subject, it is not assumed that the process is reversible.

We give sharp sufficient conditions for nonexplosions and explosions in Markov pure jump processes in terms of the holding time parameters and moments of the jump distributions.

The Hausdorff dimension and packing dimension of the image and graph of Gaussian vector fields and index-$\alpha$ stable fields are obtained under general conditions which allow for different local behavior of the components and for dependence among them. These results correct some error in print. The condition for an indexed-$\alpha$ stable field to have $k$-multiple points are considered and in the case of $Nk > (k - 1)\sum^d_{j=1} \alpha_j$, the Hausdorff dimension of the set of $k$-multiple times is given.

Let $X, X_i$ be i.i.d. real random variables with $EX^2 = \infty$. Necessary and sufficient conditions in terms of the law of $X$ are given for $(1/\gamma_n)\max_{1\leq i<j\leq n} |X_i X_j| \rightarrow 0$ a.s. in general and for $(1/\gamma_n)\sum_{1\leq i\neqj\leq n} X_i X_j \rightarrow 0$ a.s. when the variables $X_i$ are symmetric or regular and the normalizing sequence $\{\gamma_n\}$ is (mildly) regular. The rates of a.s. convergence of sums and maxima of products turn out to be different in general but to coincide under mild regularity conditions on both the law of $X$ and the sequence $\{\gamma_n\}$. Strong laws are also established for $X_{1:n} X_{k:n}$, where $X_{j:n}$ is the $j$th largest in absolute value among $X_1,\ldots,X_n$, and it is found that, under some regularity, the rate is the same for all $k \geq 3$. Sharp asymptotic bounds for $b^{-1}_n \sum^n_{i=1} X_iI_{|X_i|<b_n}$, for $b_n$ relatively small, are also obtained.

Let $E$ be a locally compact Hausdorff space with countable basis and let $(X_i)_{\i\in\mathbb{N}}$ be a family of random elements on $E$ with $(1/n) \sum^n_{i=1} \mathscr{L}(X_i) \Rightarrow^v \mu (n \rightarrow \infty)$ for a measure $\mu$ with $\|\mu\| \leq 1$. Conditions are derived under which $\mathscr{L} ((1/n) \sum^n_{i=1} \delta_{Xi}) \Rightarrow^w \delta_\mu(n \rightarrow \infty)$, where $\delta_x$ denotes the Dirac measure at $x$. The proof being based on Stein's method, there are generalisastions that allow for weak dependence between the $X_i$'s. As examples, a dissociated family and an immigration-death process are considered. The latter illustrates the possible applications in proving convergence of stochastic processes.

We prove that the set of points where exceptional oscillations of empirical and related processes occur infinitely often is a random fractal, and evaluate its Hausdorff dimension.

We give exact expansions for the upper and lower tails of the distribution of the maximum of local time of standard Brownian bridge on interval [0, 1]. We use the above expansions to prove upper and lower laws of the iterated logarithm for the maximum of the local time of the uniform empirical process. This solves two open problems cited in the book of Shorack and Wellner.

We present covariance identities and inequalities for functionals of the Wiener and the Poisson processes. Using Malliavin calculus techniques, an expansion with a remainder term is obtained for the covariance of such functionals. Our results extend known identities and inequalities for functions of multivariate random vectors.

This paper consists of three-parts. In the first-part, we find a common condition-the $C^2$ regularity--both for CLT and for moderate deviations. We show that this condition is verified in two important situations: the Lee-Yang theorem case and the FKG system case. In the second part, we apply the previous results to the additive functionals of a Markov process. By means of Feynman-Kac formula and Kasto's analytic perturbation theory, we show that the Lee-Yang theorem holds under the assumption that 1 is an isolated, simple and the only eigenvalue with modulus 1 of the operator $P_1$ acting on an appropriate Banach space $(b\mathscr{E}, C_b(E), L^2 \cdots)$. The last part is devoted to some applications to statistical mechanical systems, where the $C^2$-regularity becomes a property of the pressure functionals and the two situations presented above become exactly the Lee-Tang theorem case and the FKG system case. We shall discuss in detail the ferromagnetic model and give some general remarks on some other models.

Let $X_1, X_2,\ldots$ be independent, mean zero, uniformly bounded random variables with $S_n = X_1 + \cdots + X_n$. Optimal criteria are determined on the length and location of an interval $\Gamma$ so that $P(S_n \in \Gamma)$ is proportional to $(|\Gamma|/\sqrt{\operatorname{Var} S_n)} \wedge 1$. The proof makes an unusual use of support considerations.

Some optimal refinements of the results on strong exponential integrability of sums of independent zero-mean uniformly bounded random vectors are given.

In an earlier paper, the author proposed a universal scheme to learn the conditional distribution of the next outcome of a stationary process given the infinite past from past experience. An error in that paper is corrected.