In this paper,we establish a small time large deviation principle and obtain the following small time asymptotics:

\lim_{t \to 0}2t \log P(X_0 \in B, X_t \in C) = -d^2 (B, C),

for diffusion processes on Hilbert spaces, where $d(B,C)$ is the intrinsic metric between two subsets $B$ and $C$ associated with the diffusions. The case of perturbed Ornstein–Uhlenbeck processes is treated separately at the end of the paper.

We provide existence, comparison and stability results for one- dimensional backward stochastic differential equations (BSDEs) when the coefficient (or generator) $F(t,Y, Z)$ is continuous and has a quadratic growth in $Z$ and the terminal condition is bounded.e also give, in this framework, the links between the solutions of BSDEs set on a diffusion and viscosity or Sobolev solutions of the corresponding semilinear partial differential equations.

Consider a catalytic super-Brownian motion $X =X^\Gamma$ with finite variance branching. Here “catalytic ” means that branching of the reactant $X$ is only possible in the presence of some catalyst. Our intrinsic example of a catalyst is a stable random measure $\Gamma$ on $\mathsf{R}$ of index $0 <\gamma<1$. Consequently, here the catalyst is located in a countable dense subset of $\mathsf{R}$. Starting with a finite reactant mass $X_0$ supported by a compact set, $X$ is shown to die in finite time.We also deal with two other cases, with a power low catalyst and with a super-random walk on $\mathsf{Z^d}$ withan i.i.d.catalyst.

Our probabilistic argument uses the idea of good and bad historical paths of reactant “particles ”during time periods $[T_n, T_{n +1}$. Good paths have a signi .cant collision local time with the catalyst, and extinction can be shown by individual time change according to the collision local time and a comparison with Feller’s branching diffusion. On the other hand, the remaining bad paths are shown to have a small expected mass at time $T_{n +1}$ which can be controlled by the hitting probability of point catalysts and the collision local time spent on them.

Harmonic maps are viewed as maps sending a fixed diffusion to manifold-valued martingales.Under a convexity condition, we prove that the continuity of real-valued harmonic functions implies the continuity of harmonic maps. Then we prove with a probabilistic method that continuous harmonic maps are smooth under Hörmander’s condition; the proof relies on the study of martingales with values in the tangent bundle.

We prove that Fleming –Viot operators with parent-independent mutation satisfy a logarithmic Sobolev inequality ifand only ifthe set oftypes is finite.

This paper deals with weak convergence of stochastic integrals with respect to multivariate point processes. The results are given in terms of an entropy condition for partitioning of the index set of the integrands, which is a sort of $L^2$-bracketing.We also consider $\ell^\infty$-valued martingale difference arrays, and present natural generalizations of Jain–Marcus’s and Ossiander’s central limit theorems. As an application,the asymptotic behavior of log-likelihood ratio random fields in general statistical experiments with abstract parameters is derived.

Central limit theorems and invariance principles are obtained for additive functionals of a stationary ergodic Markov chain, say $S_n = g(X_1)+ \cdots + g(X_n)$ where $E[g(X_1)]= 0$ and $E[g(X_1)^2]<\infty$. The conditions imposed restrict the moments of $g$ and the growth of the conditional means $E(S_n|X_1)$. No other restrictions on the dependence structure of the chain are required. When specialized to shift processes,the conditions are implied by simple integral tests involving $g$.

Gardner and Derrida have introduced a natural version of the problem of the capacity of the binary perceptron “with temperature,”and they proposed (based on “physical” methods) remarkable formulas for this model. We give a complete rigorous proof that these formulas are correct at sufficiently high temperature for a much larger class of models.

Let $U$ be a multiply connected region in $\mathbf{R}^ 2$ with smooth boundary. Let $P_\epsilon$ be a polyomino in $\epsilon\mathbf{Z}^2$ approximating $U$ as $\epsilon \to 0$.We show that, for certain boundary conditions on $P_\eqsilon$, the height distribution on a random domino tiling (dimer covering) of $P_\eqsilon$ is conformally invariant in the limit as $\epsilon$ tends to 0, in the sense that the distribution of heights of boundary components (or rather, the difference of the heights from their mean values) only depends on the conformal type of $U$. The mean height is not strictly conformally invariant but transforms analytically under conformal mappings in a simple way. The mean height and all the moments are explicitly evaluated.

This paper is concerned with contact distribution functions of a random closed set $\Xi=\Bigcup_{n=1}^\infty \Xi_n$ in $\mathbb{R}^d$, where the $\Xi_n$ are assumed to be random nonempty convex bodies. These distribution functions are defined here in terms of a distance function which is associated with a strictly convex gauge body (structuring element) that contains the origin in its interior. Support measures with respect to such distances will be introduced and extended to sets in the local convex ring.These measures will then be used in a systematic way to derive and describe some of the basic properties of contact distribution functions. Most of the results are obtained in a general nonstationary setting.Only the final section deals with the stationary case.

Let $B$ be a Banach space and $\mathscr{F}$ any family of bounded linear functionals on $B$ of norm at most one. For $x\inB\set \|x\| = \sup_{\Lambda \in \mathscr{F}} \Lambda(x)(\|\cdot\|$ is at least a seminorm on $B$). We give probability estimates for the tail probability of $S_n^*= \max_{1 \leq k \leq n}\|\sum_{j=1}^{k} X_j\|$ where $\{X_i\}_{i=1}^{n}$ are independent symmetric Banach space valued random elements. Our method is based on approximating the probability that $S_n^*$ exceeds a threshold defined in terms of $\sum_{j=1}^k Y^{(j)}$, where $Y^{(r)}$ denotes the $r$th largest term of $\{\|X_i\|\}_{i=1}^n$. Using these tail estimates, essentially all the known results concerning the order of magnitude or finiteness of quantities such as $E\Phi(\|S_n|\|)$ and $E\Phi(S_n^*)$ follow (for any fixed $1 \leq n \leq \infty)$. Included in this paper are uniform $\mathscr{L}^p$ bounds of $S_n^*$ which are within a factor of 4 for all $p \geq 1$ and within a factor of 2 in the limit as $p \to \infty$.

We propose some explicit values for the constants involved in the exponential concentration inequalities for empirical processes which are due to Talagrand. It has been shown by Ledoux that deviation inequalities for empirical processes could be obtained by iteration of logarithmic Sobolev type inequalities. Our approach follows closely that of Ledoux. The improvements that we get with respect to Ledoux’s work are based on refinements of his entropy inequalities and computations.

We consider the maximum of the periodogram based on an infinite variance heavy-tailed sequence. For $\alpha < 1$ we show that the maxima constitute a weakly convergent sequence and find its limiting distribution. For $1 \leq \alpha < 2$ we show that the sequence of the maxima is not tight and find a normalization that makes it tight.

The notion of a $k$th iterated Kiefer process $\mathscr{K}(v,t;k)$ for $k \in \mathbb{N}$ and $v, t \in \mathbb{R}$ is introduced.We show that the uniform quantile process $\beta_n(t)$ may be approximated on [0,1] by $n^{-1/2} \mathscr{K}(n,t;k)$, at an optimal uniform almost sure rate of $O(n^{-1/2 + 1/2^{k+1}+o(1)})$ for each $k \in \mathbb{N}$. Our arguments are based in part on a new functional limit law, of independent interest, for the increments of the empirical process. Applications include an extended version of the uniform Bahadur–Kiefer representation, together with strong limit theorems for nonparametric functional estimators.