In this paper we show that the set of points where the density of a Wiener functional is strictly positive is an open connected set, assuming some regularity conditions.

We provide necessary and sufficient conditions for hypercontractivity of the minima of nonnegative, i.i.d. random variables and of both the maxima of minima and the minima of maxima for such r.v.'s. It turns out that the idea of hypercontractivity for minima is closely related to small ball probabilities and Gaussian correlation inequalities.

The multiplicative coalescent $X(t)$ is a $l^2$-valued Markov process representing coalescence of clusters of mass, where each pair of clusters merges at rate proportional to product of masses. From random graph asymptotics it is known (Aldous (1997)) that there exists a standard version of this process starting with infinitesimally small clusters at time $- \infty$.

In this paper, stochastic calculus techniques are used to describe all versions $(X(t);- \infty < t < \infty)$ of the multiplicative coalescent. Roughly, an extreme version is specified by translation and scale parameters, and a vector $c \in l^3$ of relative sizes of large clusters at time $- \infty$. Such a version may be characterized in three ways: via its $t \to - \infty$ behavior, via a representation of the marginal distribution $X(t)$ in terms of excursion-lengths of a Lévy-type process, or via a weak limit of processes derived from the standard version via a "coloring" construction.

Consider an embedded hypersurface $M$ in $R^3$. For $\Phi_t$ a stochastic flow of differomorphisms on $R^3$ and $x \in M$, set $x_t = \Phi_t (x)$ and $M_t = \Phi_t (M)$. In this paper we will assume $\Phi_t$ is an isotropic (to be defined below) measure preserving flow and give an explicit descripton by SDE's of the evolution of the Gauss and mean curvatures, of $M_t$ at $x_t$. If $\lambda_1 (t)$ and $\lambda_2 (t)$ are the principal curvatures of $M_t$ at $x_t$ then the vector of mean curvature and Gauss curvature, $(\lambda_1 (t) + \lambda_2 (t)$, $\lambda_1 (t) \lambda_2 (t))$, is a recurrent diffusion. Neither curvature by itself is a diffusion. In a separate addendum we treat the case of $M$ an embedded codimension one submanifold of $R^n$. In this case, there are $n-1$ principal curvatures $\lambda_1 (t), \dotsc, \lambda_{n-1} (t)$. If $P_k, k=1,\dots,n-1$ are the elementary symmetric polynomials in $\lambda_1, \dotsc, \lambda_{n-1}$, then the vector $(P_1 (\lambda_1 (t), \dotsc, \lambda_{n-1} (t)), \dotsc, P_{n-1} (\lambda_1 (t), \dotsc, \lambda_{n-1} (t))$ is a diffusion and we compute the generator explicitly. Again no projection of this diffusion onto lower dimensions is a diffusion. Our geometric study of isotropic stochastic flows is a natural offshoot of earlier works by Baxendale and Harris (1986), LeJan (1985, 1991) and Harris (1981).

Branching measure-valued diffusion models are investigated that can be regarded as pairs of historical Brownian motions modified by a competitive interaction mechanism under which individuals from each population have their longevity or fertility adversely affected by collisions with individuals from the other population. For 3 or fewer spatial dimensions, such processes are constructed using a new fixed-point technique as the unique solution of a strong equation driven by another pair of more explicitly constructible measure-valued diffusions. This existence and uniqueness is used to establish well-posedness of the related martingale problem and hence the strong Markov property for solutions. Previous work of the authors has shown that in 4 or more dimensions models with the analogous definition do not exist.

We consider a hypersurface of dimension $d$ imbedded in a $d+1$ dimensional space. For each $x\in Z^d$, let $\eta_t(x)\in R$ be the height of the surface at site $x$ at time $t$. At rate $1$ the $x$-th height is updated to a random convex combination of the heights of the `neighbors' of $x$. The distribution of the convex combination is translation invariant and does not depend on the heights. This motion, named the random average process (RAP), is one of the linear processes introduced by Liggett (1985). Special cases of RAP are a type of smoothing process (when the convex combination is deterministic) and the voter model (when the convex combination concentrates on one site chosen at random). We start the heights located on a hyperplane passing through the origin but different from the trivial one $\eta(x)\equiv 0$. We show that, when the convex combination is neither deterministic nor concentrating on one site, the variance of the height at the origin at time $t$ is proportional to the number of returns to the origin of a symmetric random walk of dimension $d$. Under mild conditions on the distribution of the random convex combination, this gives variance of the order of $t^{1/2}$ in dimension $d=1$, $\log t$ in dimension $d=2$ and bounded in $t$ in dimensions $d\ge 3$. We also show that for each initial hyperplane the process as seen from the height at the origin converges to an invariant measure on the hyper surfaces conserving the initial asymptotic slope. The height at the origin satisfies a central limit theorem. To obtain the results we use a corresponding probabilistic cellular automaton for which similar results are derived. This automaton corresponds to the product of (infinitely dimensional) independent random matrices whose rows are independent.

We present new methods for proving the convergence of the classical approximations of the Stratonovitch equation. We especially make use of the fractional Liouville-valued Sobolev space $W^{r,p}({\cal J}_{\alpha,p})$. We then obtain a support theorem for the capacity $c_{r,p}$.

We show existence of measure attractors for 2-D stochastic Navier-Stokes equations with general multiplicative noise.

Let $X$ be a Markov process with generator $A$ and let $Y(t)=\gamma (X(t))$. The conditional distribution $\pi_t$ of $X(t)$ given $\sigma (Y(s):s\leq t)$ is characterized as a solution of a filtered martingale problem. As a consequence, we obtain a generator/martingale problem version of a result of Rogers and Pitman on Markov functions. Applications include uniqueness of filtering equations, exchangeability of the state distribution of vector-valued processes, verification of quasireversibility, and uniqueness for martingale problems for measure-valued processes. New results on the uniqueness of forward equations, needed in the proof of uniqueness for the filtered martingale problem are also presented.

We consider critical site percolation on the triangular lattice, that is, we choose $X(v) = 0$ or 1 with probability 1/2 each, independently for all vertices $v$ of the triangular lattice. We say that a word $(\xi_1, \xi_2,\dots) \in \{0,1\}^{\Bbb N}$ is seen in the percolation configuration if there exists a selfavoiding path $(v_1, v_2, \dots)$ on the triangular lattice with $X(v_i) = \xi_i, i \ge 1$. We prove that with probability 1 "almost all" words, as well as all periodic words, except the two words $(1,1,1, \dots)$ and $(0,0,0,\dots)$, are seen. "Almost all" words here means almost all with respect to the measure $\mu_\beta$ under which the $\xi_i$ are i.i.d. with $\mu_\beta {\xi_i = 0}=1 - \mu_\beta {\xi_i = 1} = \beta$ (for an arbitrary $0 <\beta < 1$).

The Cauchy problem for 1-dimensional nonlinear stochastic partial differential equations is studied. The uniqueness and existence of solutions in $cH^2_p(T)$-space are proved.

Let $X$ and $Y$ be time-homogeneous Markov processes with common state space $E$, and assume that the transition kernels of $X$ and $Y$ admit densities with respect to suitable reference measures. We show that if there is a time $t>0$ such that, for each $x\in E$, the conditional distribution of $(X_s)_{0\le s\le t}$, given $X_0=x=X_t$, coincides with the conditional distribution of $(Y_s)_{0\le s\le t}$, given $Y_0=x=Y_t$, then the infinitesimal generators of $X$ and $Y$ are related by $L^Yf=\psi^{-1}L^X(\psi f)-\lambda f$, where $\psi$ is an eigenfunction of $L^X$ with eigenvalue $\lambda\in{\bf R}$. Under an additional continuity hypothesis, the same conclusion obtains assuming merely that $X$ and $Y$ share a "bridge" law for one triple $(x,t,y)$. Our work extends and clarifies a recent result of I. Benjamini and S. Lee.

We obtain a Laplace asymptotic expansion, in orders of $\lambda$, of $$ E^\rho_x \left\{ G(\lambda x) e^{-\lambda ^{-2} F(\lambda x)}\right\}$$ the expectation being with respect to a Gaussian process. We extend a result of Pincus and build upon the previous work of Davies and Truman. Our methods differ from those of Ellis and Rosen in that we use the supremum norm to simplify the application of the result.

Let $U(n)$ denote the most visited point by a simple symmetric random walk $\{ S_k\}_{k\ge 0}$ in the first $n$ steps. It is known that $U(n)$ and $max_{0\le k\le n} S_k$ satisfy the same law of the iterated logarithm, but have different upper functions (in the sense of P. Lévy). The distance between them however turns out to be transient. In this paper, we establish the exact rate of escape of this distance. The corresponding problem for the Wiener process is also studied.

Let $(f_n)$ be a mean zero vector valued martingale sequence. Then there exist vector valued functions $(d_n)$ from $[0,1]^n$ such that $\int_0^1 d_n(x_1,\dots,x_n)\,dx_n = 0$ for almost all $x_1,\dots,x_{n-1}$, and such that the law of $(f_n)$ is the same as the law of $(\sum_{k=1}^n d_k(x_1,\dots,x_k))$. Similar results for tangent sequences and sequences satisfying condition (C.I.) are presented. We also present a weaker version of a result of McConnell that provides a Skorohod like representation for vector valued martingales.