Consider diffusions on $\mathbb{R}^k > 1$, governed by the Itô equation $dX(t) = {b(X(t)) + \beta(X(t)/a)} dt + \sigmadB(t)$, where $b, \beta$ are periodic with the same period and are divergence free, $\sigma$ is nonsingular and a is a large integer. Two distinct Gaussian phases occur as time progresses. The initial phase is exhibited over times $1 \ll t \ll a^{2/3}$. Under a geometric condition on the velocity field $\beta$, the final Gaussian phase occurs for times $t \gg a^2(\log a)^2$, and the dispersion grows quadratically with a . Under a complementary condition, the final phase shows up at times $t \gg a^4(\log a)^2$, or $t \gg a^2 \log a$ under additional conditions, with no unbounded growth in dispersion as a function of scale. Examples show the existence of non-Gaussian intermediate phases. These probabilisitic results are applied to analyze a multiscale Fokker-Planck equation governing solute transport in periodic porous media. In case $b, \beta$ are not divergence free, some insight is provided by the analysis of one-dimensional multiscale diffusions with periodic coefficients.

We introduce two new types of random field. The cross correlation field $R(\mathbf{s}, \mathbf{t})$ is the usual sample correlation coefficient for a set of pairs of Gaussian random fields, one sampled at point $s \epsilon \Re^M$, the other sampled at point $\mathbf{t} \epsilon \Re^N$. The homologous correlation field is defined as $R(\mathbf{t}) = R(\mathbf{t}, \mathbf{t})$, that is, the "diagonal" of the cross correlation field restricted to the same location $\mathbf{s} = \mathbf{t}$. Although the correlation coefficient can be transformed pointwise to a t-statistic, neither of the two correlation fields defined above can be transformed to a t-field, defined as a standard Gaussian field divided by the root mean square of i.i.d. standard Gaussian fields. For this reason, new results are derived for the geometry of the excursion set of these correlation fields that extend those of Adler. The results are used to detect functional connectivity (regions of high correlation) in three-dimensional positron emission tomography (PET) images of human brain activity.

We develop the filtering theory in the case where both the signal and the observation are solutions of some stochastic differential equation driven by a multidimensional fractional Brownian motion. We show that the classical approach fails to give a closed equation for the filter and we develop another approach using an auxiliary process-valued semimartingale which solves this problem theoretically.

Infinite population genetic models with general type space incorporating mutation, selection and recombination are considered. The Fleming-Viot measure-valued diffusion is represented in terms of a countably infinite-dimensional process. The complete genealogy of the population at each time can be recovered from the model. Results are given concerning the existence of stationary distributions and ergodicity and absolute continuity of the stationary distribution for a model with selection with respect to the stationary distribution for the corresponding neutral model.

The fuzzy Potts model arises by taking the q-state Potts model, then identifying r of the Potts spins with the fuzzy spin 0, and the remaining $q - r$ Potts spins with the fuzzy spin 1. Here we extend a result of Chayes by showing that the fuzzy Potts model has positive correlations. We also give an application to the percolation-theoretic behavior of the Potts model on $\mathbf{Z}^2$.

A (general) branching process, where individuals need not reproduce independently, satisfies a homogeneous growth condition if, vaguely, one would not expect the progeny from any one individual to make out more than its proper fraction of the whole population at any time in the future. This notion is made precise, and it is shown how it entails classical Malthusian growth in supercritical cases, in particular for population size-dependent Bienaymé-Galton-Watson and Markov branching processes, and for nondecreasing age-dependent processes with continuous life span distributions.

Let $\mathscr{D} \subseteq (-\infty, \infty)$ be closed domain and set $\xi = \inf{x;x \epsilon \mathscr{D}}$. Let the sequence $\mathscr{X}^n = {X_j^{(n)}; j \geq 1}, n \geq 1$ be associated with the sequence of measurable iterated functions $f_n(x_1, x_2,\dots, x_{k_n}): \mathscr{D}^{k_n} \rightarrow \mathscr{D} (k_n \geq 2), n \geq 1$ and some initial sequence $\mathscr{X}^{(0)} = {X_j^{(0)}; j \geq 1}$ of stationary and m-dependent random variables such that $P(X_1^{(0)} \epsilon \mathscr{D}) = 1$ and $X_j^{(n)} = f_n(X_{(j-1)k_n+1}^{(n-1)},\dots, X_{jk_n}^{(n-1)}), j \geq 1, n \geq 1$. This paper studies the asymptotic behavior for the hierarchical sequence ${X_1^{(n)}; n \geq 0}$. We establish general asymptotic results for such sequences under some surprisingly relaxed conditions. Suppose that, for each $n \geq 1$, there exist $k_n$ non-negative constants $\alpha_{n, i}, 1 \leq i \leq k_n$ such that $\Sigma_{i=1}^{k_n} \alpha_{n, i} = 1$ and $f_n(x_1,\dots, x_{k_n}) \leq \Sigma_{i=1}^{k_n} \alpha_{n, i}x_i, \forall(x_1,\dots, x_{k_n}) \epsilon \mathscr{D}^{k_n}$. If $\Pi_{j=1}^n \max_{1\leqi\leqk_j \alpha_{j, i} \rightarrow 0$ as $n \rightarrow \infty$ and $E(X_1^{(n)}) \downarrow \lambda$ as $n \rightarrow \infty$ and $X_1^{(n)} \rightarrow_P \lambda$. We conclude with various examples, comments and open questions and discuss further how our results can be applied to models arising in mathematical physics.

This paper analyzes the performance of importance sampling distributions for computing expectations with respect to a whole family of probability laws in the context of Markov chain Monte Carlo simulation methods. Motivations for such a study arise in statistics as well as in statistical physics. Two choices of importance sampling distributions are considered in detail: mixtures of the distributions of interest and distributions that are "uniform over energy levels" (motivated by physical applications). We analyze two examples, a "witch's hat" distribution and the mean field Ising model, to illustrate the advantages that such simulation procedures are expected to offer in a greater generality. The connection with the recently proposed simulated tempering method is also examined.

We consider a spatial stochastic version of the classical Lotka-Volterra model with interspecific competition.

The classical model is described by a set of ordinary differential equations, one for each species. Mortality is density dependent, including both intraspecific and interspecific competition. Fecundity may depend on the type of species but is density independent. Depending on the relative strengths of interspecific and intraspecific competition and on the fecundities, the parameter space for the classical model is divided into regions where either coexistence, competitive exclusion or founder control occur.

The spatial version is a continuous time Markov process in which individuals are located on the d-dimensional integer lattice. Their dynamics are described by a set of local rules which have the same components as the classical model.

Our main results for the spatial stochastic version can be summarized as follows. Local competitive interactions between species result in (1) a reduction of the parameter region where coexistence occurs in the classical model, (2) a reduction of the parameter region where founder control occurs in the classical model, and (3) spatial segregation of the two species in parts of the parameter region where the classical model predicts coexistence.

For various global characteristics of large size combinatorial structures, such as graphs, trees, one can usually estimate the mean and the variance, and also obtain a recurrence for the generating function, with the structure size n serving as the recursive parameter. As a heuristic principle based on our experience, we claim that such a characteristic is asymptotically normal if the mean and the variance are "nearly linear" in n. The technical reason is that in such a case the moment generating function (the characteristic function) of the normal distribution with the same two moments "almost" satisfies the recurrence. Of course, an actual proof may well depend on a magnitude of the relative error, and the latter is basically determined by degree of nonlinearity of the mean and the variance. We provide some new illustrations of this paradigm. The uniformly random tree on n-labelled vertices is studied. Using and strengthening the earlier results of Meir and Moon, we show that the independence number is asymptoticaly normal, with mean $\rhon$ and variance $\sigma^2n, (\sigma^2 = \rho(1-\rho-\rho^2)(1+\rho)^{-1})$; here $\rho \approx 0.5671$ is the root of $xe^x = 1$. As a second example, we prove that--in the rooted tree--the counts of vertices with total progeny of various sizes form an asymptotically Gaussian sequence. This is an extension of Rényi's result on asymptotic normality of the number of leaves in the random tree. In both cases we establish convergence of the generating function. Finally we show that the overall number of ways to extend, totally, the tree-induced partial order is lognormal in the limit, with mean and variance roughly $\logn!-an$ and $bn \log n$. The proof is based on convergence of the cumulants of the generating function.

In the framework of continuous-time, Itô processes models for financial markets, we study the problem of maximizing the probability of an agent's wealth at time T being no less than the value C of a contingent claim with expiration time T. The solution to the problem has been known in the context of complete markets and recently also for incomplete markets; we rederive the complete markets solution using a powerful and simple duality method, developed in utility maximization literature. We then show how to modify this approach to solve the problem in a market with partial information, the one in which we have only a prior distribution on the vector of return rates of the risky assets. Finally, the same problem is solved in markets in which the wealth process of the agent has a nonlinear drift. These include the case of different borrowing and lending rates, as well as "large investor" models. We also provide a number of explicitly solved examples.