We extend J.B. Walsh's theory of martingale measures in order to deal with stochastic partial differential equations that are second order in time, such as the wave equation and the beam equation, and driven by spatially homogeneous Gaussian noise. For such equations, the fundamental solution can be a distribution in the sense of Schwartz, which appears as an integrand in the reformulation of the s.p.d.e. as a stochastic integral equation. Our approach provides an alternative to the Hilbert space integrals of Hilbert-Schmidt operators. We give several examples, including the beam equation and the wave equation, with nonlinear multiplicative noise terms.

In this paper we give a solution for the one-dimensional reflected backward stochastic differential equation when the noise is driven by a Brownian motion and an independent Poisson point process. We prove existence and uniqueness of the solution in using penalization and the Snell envelope theory. However both methods use a contraction in order to establish the result in the general case. Finally, we highlight the connection of such reflected BSDEs with integro-differential mixed stochastic optimal control.

The classical stationary Ornstein-Uhlenbeck process can be obtained in two different ways. On the one hand, it is a stationary solution of the Langevin equation with Brownian motion noise. On the other hand, it can be obtained from Brownian motion by the so called Lamperti transformation. We show that the Langevin equation with fractional Brownian motion noise also has a stationary solution and that the decay of its auto-covariance function is like that of a power function. Contrary to that, the stationary process obtained from fractional Brownian motion by the Lamperti transformation has an auto-covariance function that decays exponentially.

The paper focuses on spatial multitype branching systems with spatial components (colonies) indexed by a countable group, for example $Z^d$ or the hierarchical group. As type space we allow continua and describe populations in one colony as measures on the type space. The spatial components of the system interact via migration. Instead of the classical independence assumption on the evolution of different families of the branching population, we introduce interaction between the families through a state dependent branching rate of individuals and in addition state dependent mean offspring of individuals. However for most results we consider the critical case in this work. The systems considered arise as diffusion limits of critical multiple type branching random walks on a countable group with interaction between individual families induced by a branching rate and offspring mean for a single particle, which depends on the total population at the site at which the particle in question is located.

The main purpose of this paper is to construct the measure valued diffusions in question, characterize them via well-posed martingale problems and finally determine their longtime behavior, which includes some new features. Furthermore we determine the dynamics of two functionals of the system, namely the process of total masses at the sites and the relative weights of the different types in the colonies as system of interacting diffusions respectively time-inhomogeneous Fleming-Viot processes. This requires a detailed analysis of path properties of the total mass processes.

In addition to the above mentioned systems of interacting measure valued processes we construct the corresponding historical processes via well-posed martingale problems. Historical processes include information on the family structure, that is, the varying degrees of relationship between individuals.

Ergodic theorems are proved in the critical case for both the process and the historical process as well as the corresponding total mass and relative weights functionals. The longtime behavior differs qualitatively in the cases in which the symmetrized motion is recurrent respectively transient. We see local extinction in one case and honest equilibria in the other.

This whole program requires the development of some new techniques, which should be of interest in a wider context. Such tools are dual processes in randomly fluctuating medium with singularities and coupling for systems with multi-dimensional components.

The results above are the basis for the analysis of the large space-time scale behavior of such branching systems with interaction carried out in a forthcoming paper. In particular we study there the universality properties of the longtime behavior and of the family (or genealogical) structure, when viewed on large space and time scales.

Shnerb et al. (2000), (2001) studied the following system of interacting particles on $\Bbb Z^d$: There are two kinds of particles, called $A$-particles and $B$-particles. The $A$-particles perform continuous time simple random walks, independently of each other. The jumprate of each $A$-particle is $D_A$. The $B$-particles perform continuous time simple random walks with jumprate $D_B$, but in addition they die at rate $\delta$ and a $B$-particle at $x$ at time $s$ splits into two particles at $x$ during the next $ds$ time units with a probability $\beta N_A(x,s)ds +o(ds)$, where $N_A(x,s)\; (N_B(x,s))$ denotes the number of $A$-particles (respectively $B$-particles) at $x$ at time $s$. Conditionally on the $A$-system, the jumps, deaths and splittings of different $B$-particles are independent. Thus the $B$-particles perform a branching random walk, but with a birth rate of new particles which is proportional to the number of $A$-particles which coincide with the appropriate $B$-particles. One starts the process with all the $N_A(x,0),\, x \in \Bbb Z^d$, as independent Poisson variables with mean $\mu_A$, and the $N_B(x,0),\, x \in \Bbb Z^d$, independent of the $A$-system, translation invariant and with mean $\mu_B$.

Shnerb et al. (2000) made the interesting discovery that in dimension 1 and 2 the expectation $\Bbb E\{N_B(x,t)\}$ tends to infinity, no matter what the values of $\delta, \beta, D_A$, $D_B, \mu_A,\mu_B \in (0, \infty)$ are. We shall show here that nevertheless there is a phase transition in all dimensions, that is, the system becomes (locally) extinct for large $\delta$ but it survives for $\beta$ large and $\delta$ small.

We consider the stochastic heat equation with a multiplicative colored noise term on the real space for dimensions greater or equal to 1. First, we prove convergence of a branching particle system in a random environment to this stochastic heat equation with linear noise coefficients. For this stochastic partial differential equation with more general non-Lipschitz noise coefficients we show convergence of associated lattice systems, which are infinite dimensional stochastic differential equations with correlated noise terms, provided that uniqueness of the limit is known. In the course of the proof, we establish existence and uniqueness of solutions to the lattice systems, as well as a new existence result for solutions to the stochastic heat equation. The latter are shown to be jointly continuous in time and space under some mild additional assumptions.

Given a real or complex $n \times n$ matrix $A_n$, we compute the expected value and the variance of the random variable $\| A_n x\|^2/\| A_n \|^2$, where $x$ is uniformly distributed on the unit sphere of $R^n$ or $C^n$. The result is applied to several classes of structured matrices. It is in particular shown that if $A_n$ is a Toeplitz matrix $T_n(b)$, then for large $n$ the values of $\| A_n x\|/\| A_n \|$ cluster fairly sharply around $\| b \|_2/\| b \|_\infty$ if $b$ is bounded and around zero in case $b$ is unbounded.

We study pairs of interacting measure-valued branching processes (superprocesses) with alpha-stable migration and $(1+\beta)$-branching mechanism. The interaction is realized via some killing procedure. The collision local time for such processes is constructed as a limit of approximating collision local times. For certain dimensions this convergence holds uniformly over all pairs of such interacting superprocesses. We use this uniformity to prove existence of a solution to a competing species martingale problem under a natural dimension restriction. The competing species model describes the evolution of two populations where individuals of different types may kill each other if they collide. In the case of Brownian migration and finite variance branching, the model was introduced by Evans and Perkins (1994). The fact that now the branching mechanism does not have finite variance requires the development of new methods for handling the collision local time which we believe are of some independent interest.

We derive the optimal convergence rate $O(n^{-1/4})$ in the central limit theorem for the number of maxima in random samples chosen uniformly at random from the right equilateral triangle with two sides parallel to the axes, the hypotenuse with the slope $-1$ and consituting the top part of the boundary of the triangle. A local limit theorem with rate is also derived. The result is then applied to the number of maxima in general planar regions (upper-bounded by some smooth decreasing curves) for which a near-optimal convergence rate to the normal distribution is established.

The potential kernel of a positive left additive functional (of a strong Markov process $X$) maps positive functions to strongly supermedian functions and satisfies a variant of the classical domination principle of potential theory. Such a kernel $V$ is called a regular strongly supermedian kernel in recent work of L. Beznea and N. Boboc. We establish the converse: Every regular strongly supermedian kernel $V$ is the potential kernel of a random measure homogeneous on $[0,\infty[$. Under additional finiteness conditions such random measures give rise to left additive functionals. We investigate such random measures, their potential kernels, and their associated characteristic measures. Given a left additive functional $A$ (not necessarily continuous), we give an explicit construction of a simple Markov process $Z$ whose resolvent has initial kernel equal to the potential kernel $U_{\!A}$. The theory we develop is the probabilistic counterpart of the work of Beznea and Boboc. Our main tool is the Kuznetsov process associated with $X$ and a given excessive measure $m$.

Clustering behavior is studied for a continuous-sites stepping-stone model with Brownian migration. It is shown that, if the model starts with the same mixture of different types of individuals over each site, then it will evolve in a way such that the site space is divided into disjoint intervals where only one type of individuals appear in each interval. Those intervals (clusters) are growing as time $t$ goes to infinity. The average size of the clusters at a fixed time $t$ is of the order of square root of $t$. Clusters at different times or sites are asymptotically independent as the difference of either the times or the sites goes to infinity.

In this paper we prove a large deviation principle (LDP) for a perturbed stochastic heat equation defined on $[0,T]\times [0,1]^d$. This equation is driven by a Gaussian noise, white in time and correlated in space. Firstly, we show the Holder continuity for the solution of the stochastic heat equation. Secondly, we check that our Gaussian process satisfies an LDP and some requirements on the skeleton of the solution. Finally, we prove the called Freidlin-Wentzell inequality. In order to obtain all these results we need precise estimates of the fundamental solution of this equation.

In this paper we consider the Laplace transforms of some random series, in particular, the random series derived as the squared $L_2$ norm of a Gaussian stochastic process. Except for some special cases, closed form expressions for Laplace transforms are, in general, rarely obtained. It is the purpose of this paper to show that for many Gaussian random processes the Laplace transform can be expressed in terms of well understood functions using complex-analytic theorems on infinite products, in particular, the Hadamard Factorization Theorem. Together with some tools from linear differential operators, we show that in many cases the Laplace transforms can be obtained with little work. We demonstrate this on several examples. Of course, once the Laplace transform is known explicitly one can easily calculate the corresponding exact $L_2$ small ball probabilities using Sytaja Tauberian theorem. Some generalizations are mentioned.

Using the techniques of the stochastic calculus of variations for Gaussian processes, we derive an Itô formula for the fractional Brownian sheet with Hurst parameters bigger than $1/2$. As an application, we give a stochastic integral representation for the local time of the fractional Brownian sheet.

Let $M$ be a smooth, compact, connected Riemannian manifold of dimension $d \gt 2$ and without boundary. Denote by $T(x,r)$ the hitting time of the ball of radius $r$ centered at $x$ by Brownian motion on $M$. Then, $C_r(M)=\sup_{x \in M} T(x,r)$ is the time it takes Brownian motion to come within $r$ of all points in $M$. We prove that $C_r(M)/(r^{2-d}|\log r|)$ tends to $\gamma_d V(M)$ almost surely as $r\to 0$, where $V(M)$ is the Riemannian volume of $M$. We also obtain the ``multi-fractal spectrum'' $f(\alpha)$ for ``late points'', i.e., the dimension of the set of $\alpha$-late points $x$ in $M$ for which $\limsup_{r\to 0} T(x,r)/ (r^{2-d}|\log r|) = \alpha \gt 0$.

We consider the large deviations properties of the empirical measure for one dimensional constrained processes, such as reflecting Brownian motion, the M/M/1 queue, and discrete time analogues. Because these processes do not satisfy the strong stability assumptions that are usually assumed when studying the empirical measure, there is significant probability (from the perspective of large deviations) that the empirical measure charges the point at infinity. We prove the large deviation principle and identify the rate function for the empirical measure for these processes. No assumption of any kind is made with regard to the stability of the underlying process.

We consider in this paper the support $[L',R']$ of the one dimensional Integrated Super Brownian Excursion. We determine the distribution of $(R',L')$ through a modified Laplace transform. Then we give an explicit value for the first two moments of $R'$ as well as the covariance of $R'$ and ${L'}$.

Let $X$ be the fractional Brownian motion of any Hurst index $H\in (0,1)$ (resp. a semimartingale) and set $\alpha=H$ (resp. $\alpha=\frac{1}{2}$). If $Y$ is a continuous process and if $m$ is a positive integer, we study the existence of the limit, as $\varepsilon\rightarrow 0$, of the approximations $$ I_{\varepsilon}(Y,X) :=\left\{\int_{0}^{t}Y_{s}\left(\frac{X_{s+\varepsilon}-X_{s}}{\varepsilon^{\alpha}}\right)^{m}ds,\,t\geq 0\right\} $$ of $m$-order integral of $Y$ with respect to $X$. For these two choices of $X$, we prove that the limits are almost sure, uniformly on each compact interval, and are in terms of the $m$-th moment of the Gaussian standard random variable. In particular, if $m$ is an odd integer, the limit equals to zero. In this case, the convergence in distribution, as $\varepsilon\rightarrow 0$, of $\varepsilon^{-\frac{1}{2}} I_{\varepsilon}(1,X)$ is studied. We prove that the limit is a Brownian motion when $X$ is the fractional Brownian motion of index $H\in (0,\frac{1}{2}]$, and it is in term of a two dimensional standard Brownian motion when $X$ is a semimartingale.

The solution of the Langevin equation driven by a Lévy process noise has been well studied, under the name of Ornstein-Uhlenbeck type process. It is a stationary Markov process. When the noise is fractional Brownian motion, the covariance of the stationary solution process has been studied by the first author with different coauthors. In the present paper, we consider the Langevin equation driven by a linear fractional stable motion noise, which is a selfsimilar process with long-range dependence but does not have finite variance, and we investigate the dependence structure of the solution process.

The higher-order heat-type equation $ \partial u/\partial t=\pm\partial^{n} u/ \partial x^{n} $ has been investigated by many authors. With this equation is associated a pseudo-process $(X_t)_{t\ge 0}$ which is governed by a signed measure. In the even-order case, Krylov (1960) proved that the classical arc-sine law of Paul Levy for standard Brownian motion holds for the pseudo-process $(X_t)_{t\ge 0}$, that is, if $T_t$ is the sojourn time of $(X_t)_{t\ge 0}$ in the half line $(0,+\infty)$ up to time $t$, then $P(T_t\in ds)=\frac{ds}{\pi\sqrt{s(t-s)}}$, $0 \lt s \lt t$. Orsingher (1991) and next Hochberg and Orsingher (1994) obtained a counterpart to that law in the odd cases $n=3,5,7.$ Actually Hochberg and Orsingher (1994) proposed a more or less explicit expression for that new law in the odd-order general case and conjectured a quite simple formula for it. The distribution of $T_t$ subject to some conditioning has also been studied by Nikitin & Orsingher (2000) in the cases $n=3,4.$ In this paper, we prove that the conjecture of Hochberg and Orsingher (1994) is true and we extend the results of Nikitin & Orsingher for any integer $n$. We also investigate the distributions of maximal and minimal functionals of $(X_t)_{t\ge 0}$, as well as the distribution of the last time before becoming definitively negative up to time $t$.

Let ${\xi_n}$ be a sequence of i.i.d. positive random variables with common distribution function $F(x)$. Let ${a_n}$ and ${b_n}$ be two positive non-increasing summable sequences such that ${\prod_{n=1}^{\infty}(a_n/b_n)}$ converges. Under some mild assumptions on $F$, we prove the following comparison $$P\left(\sum_{n=1}^{\infty}a_n \xi_n \leq \varepsilon \right) \sim \left(\prod_{n=1}^{\infty}\frac{b_n}{a_n}\right)^{-\alpha} P \left(\sum_{n=1}^{\infty}b_n \xi_n \leq \varepsilon \right),$$ where $${ \alpha=\lim_{x\to \infty}\frac{\log F(1/x)}{\log x}} \lt 0$$ is the index of variation of $F(1/\cdot)$. When applied to the case $\xi_n=|Z_n|^p$, where $Z_n$ are independent standard Gaussian random variables, it affirms a conjecture of Li (1992).

This paper studies the convergence rate of solutions of the linear Itô-Volterra equation $$ dX(t) = \left(AX(t) + \int_{0}^{t} K(t-s)X(s),ds\right)\,dt + \Sigma(t)\,dW(t) \tag{1} $$ where $K$ and $\Sigma$ are continuous matrix-valued functions defined on $\mathbb{R}^{+}$, and $(W(t))_{t \geq 0}$ is a finite-dimensional standard Brownian motion. It is shown that when the entries of $K$ are all of one sign on $\mathbb{R}^{+}$, that (i) the almost sure exponential convergence of the solution to zero, (ii) the $p$-th mean exponential convergence of the solution to zero (for all $p \gt 0$), and (iii) the exponential integrability of the entries of the kernel $K$, the exponential square integrability of the entries of noise term $\Sigma$, and the uniform asymptotic stability of the solutions of the deterministic version of (1) are equivalent. The paper extends a result of Murakami which relates to the deterministic version of this problem.

We consider a nearest-neighbor stochastic process on a rooted tree $G$ which goes toward the root with probability $1-\varepsilon$ when it visits a vertex for the first time. At all other times it behaves like a simple random walk on $G$. We show that for all $\varepsilon\ge 0$ this process is transient. Also we consider a generalization of this process and establish its transience in some cases.

We study the problem of stationary control by adaptive choice of the diffusion coefficient in the case that control degeneracy is allowed and the drift admits a unique, asymptotically stable equilibrium point. We characterize the optimal value and obtain it as an Abelian limit of optimal discounted values and as a limiting average of finite horizon optimal values, and we also characterize the optimal stationary strategy. In the case of linear drift, the optimal stationary value is expressed in terms of the solution of an optimal stopping problem. We generalize the above results to allow unbounded cost functions.