Let $D$ be a bounded domain in the plane which is symmetric and convex with respect to both coordinate axes. We prove that the Brownian motion conditioned to remain forever in $D$, the Doob $h$-process where $h$ is the ground state Dirichlet eigenfunction in $D$, has the "hot-spots" property. That is, the first non-constant eigenfunction corresponding to the semigroup of this process with its nodal line on one of the coordinate axes attains its maximum and minimum on the boundary and only on the boundary of the domain. This is the exact analogue for conditioned Brownian motion of the result in \cite{JN} for Neumann eigenfunctions.

Let $\mathcal{G}$ be the incipient infinite cluster (IIC) for percolation on a homogeneous tree of degree $n_0 + 1$. We obtain estimates for the transition density of the continuous time simple random walk $Y$ on $\mathcal{G}$; the process satisfies anomalous diffusion and has spectral dimension 4/3.

Importance sampling, particularly sequential and adaptive importance sampling, have emerged as competitive simulation techniques to Markov-chain Monte-Carlo techniques. We compare importance sampling and the Metropolis algorithm as two ways of changing the output of a Markov chain to get a different stationary distribution.

We give a geometric derivation of $\text{SLE}(\kappa,\rho)$ in terms of conformally invariant random growing compact subsets of polygons. The parameters $\rho_j$ are related to the exterior angles of the polygons. We also show that $\text{SLE}(\kappa,\rho)$ can be generated by a metric Brownian motion, where metric and Brownian motion are coupled and the metric is a pull-back metric of the Euclidean metric of an evolving polygon.

We prove a law of the iterated logarithm for the Kolmogorov-Smirnov statistic, or equivalently, the discrepancy of sequences $(n_{k}\omega)$ mod $1$. Here $(n_{k})$ is a sequence of integers satisfying a sub-Hadamard growth condition and such that linear Diophantine equations in the variables $n_{k}$ do not have too many solutions. The proof depends on a martingale embedding of the empirical process; the number-theoretic structure of $(n_k)$ enters through the behavior of the square function of the martingale.

We prove several limit theorems that relate coalescent processes to continuous-state branching processes. Some of these theorems are stated in terms of the so-called generalized Fleming-Viot processes, which describe the evolution of a population with fixed size, and are duals to the coalescents with multiple collisions studied by Pitman and others. We first discuss asymptotics when the initial size of the population tends to infinity. In that setting, under appropriate hypotheses, we show that a rescaled version of the generalized Fleming-Viot process converges weakly to a continuous-state branching process. As a corollary, we get a hydrodynamic limit for certain sequences of coalescents with multiple collisions: Under an appropriate scaling, the empirical measure associated with sizes of the blocks converges to a (deterministic) limit which solves a generalized form of Smoluchowski's coagulation equation. We also study the behavior in small time of a fixed coalescent with multiple collisions, under a regular variation assumption on the tail of the measure $\nu$ governing the coalescence events. Precisely, we prove that the number of blocks with size less than $\varepsilon x$ at time $(\varepsilon\nu([\varepsilon,1]))^{-1}$ behaves like $\varepsilon^{-1}\lambda_1(]0,x[)$ as $\varepsilon\to 0$, where $\lambda_1$ is the distribution of the size of one cluster at time $1$ in a continuous-state branching process with stable branching mechanism. This generalizes a classical result for the Kingman coalescent.

In this paper we describe some recent developments on the problems of existence of quasi-periodic and almost-periodic solutions and diffusion for nonlinear Schrödinger equations with a random potential on the lattice and in the continuum.

For every bounded planar domain $D$ with a smooth boundary, we define a ``Lyapunov exponent'' $\Lambda(D)$ using a fairly explicit formula. We consider two reflected Brownian motions in $D$, driven by the same Brownian motion (i.e., a ``synchronous coupling''). If $\Lambda(D)>0$ then the distance between the two Brownian particles goes to $0$ exponentially fast with rate $\Lambda (D)/(2|D|)$ as time goes to infinity. The exponent $\Lambda(D)$ is strictly positive if the domain has at most one hole. It is an open problem whether there exists a domain with $\Lambda(D)<0$.

Let $(X, \wh X)$ be a pair of Borel standard processes on a Lusin space $E$ that are in weak duality with respect to some $\sigma$-finite measure $m$ that has full support on $E$. Let $F$ be a finely closed subset of $E$. In this paper, we obtain the characterization of a L\'evy system of the time changed process of $X$ by a positive continuous additive functional (PCAF in abbreviation) of $X$ having support $F$, under the assumption that every $m$-semipolar set of $X$ is $m$-polar for $X$. The characterization of the L\'evy system is in terms of Feller measures, which are intrinsic quantities for the part process of $X$ killed upon leaving $E\setminus F$. Along the way, various relations between the entrance law, exit system, Feller measures and the distribution of the starting and ending point of excursions of $X$ away from $F$ are studied. We also show that the time changed process of $X$ is a special standard process having a weak dual and that the $\mu$-semipolar set of $Y$ is $\mu$-polar for $Y$, where $\mu$ is the Revuz measure for the PCAF used in the time change.

For $0< p <\infty$, put

\[ Y_t(c,p)=Y= B_t^{*(p-2)} [ B_t^2 -t ]+c B_t^{*p},\quad t>0, \]

where $B_t$ is a Brownian Motion and $B_t^*=\max_{0 \leq s \leq t} |B_s|$. Then for $0< p \leq 2$, $Y$ is a submartingale if and only if $c \geq \frac{2-p}{p}$, while for $2 \leq p < \infty$, $Y$ is a supermartingale if and only if $c\leq \frac{2-p}{p}$. This extends results of Burkholder. The first of these assertions implies a strong version of some of the Burkholder-Gundy inequalities.

Weak uniqueness is established for the martingale problem associated to a family of catalytic branching networks. This martingale problem corresponds to a stochastic differential equation with a degenerate Hölder continuous diffusion matrix. Our approach uses the semigroup perturbation method of Stroock and Varadhan and a modification of a Banach space of weighted Hölder continuous functions introduced by Bass and Perkins.

The Martin boundary theory allows one to describe all positive harmonic functions in an arbitrary domain $E$ of a Euclidean space starting from the functions $k^y(x)=\sfrac{g(x,y)}{g(a,y)}$, where $g(x,y)$ is the Green function of the Laplacian and $a$ is a fixed point of $E$. In two previous papers a similar theory was developed for a class of positive functions on a space of measures. These functions are associated with a superdiffusion $X$ and we call them $X$-harmonic. Denote by $\M_c(E)$ the set of all finite measures $\mu$ supported by compact subsets of $E$. $X$-harmonic functions are functions on $\M_c(E)$ characterized by a mean value property formulated in terms of exit measures of a superdiffusion. Instead of the ratio $\sfrac{g(x,y)}{g(a,y)}$ we use a Radon-Nikodym derivative of the probability distribution of an exit measure of $X$ with respect to the probability distribution of another such measure. The goal of the present note is to find an expression for this derivative.

An open problem in martingale theory is to characterize the normal martingales having the chaotic representation property. This property is shown to hold for certain families of multidimensional Azéma martingales.

Excursions from a fixed point $b$ are studied in the framework of a general Borel right process $X$, with a fixed excessive measure $m$ serving as background measure; such a measure always exists if $b$ is accessible from every point of the state space of $X$. In this context the left-continuous moderate Markov dual process $\widehat X$ arises naturally and plays an important role. This allows the basic quantities of excursion theory such as the Laplace-L\'evy exponent of the inverse local time at $b$ and the Laplace transform of the entrance law for the excursion process to be expressed as inner products involving simple hitting probabilities and expectations. In particular if $X$ and $\widehat X$ are honest, then the resolvent of $X$ may be expressed entirely in terms of quantities that depend only on $X$ and $\widehat X$ killed when they first hit $b$.

Given a convex function $f$ and a set $\Q$ of probability measures, we consider the problem of minimizing the robust $f$-divergence $\infq f(P|Q)$ over the class $\PP$ of martingale measures. Under mild conditions on $\PP$ and $\Q$ we show that a minimizer exists within the class $\PP$ if $\lim_{x \rightarrow \infty} f(x)/x = \infty$. If $\lim_{x \rightarrow \infty} f(x)/x = 0$ then there is a minimizer in a class $\bar\PP$ of extended martingale measures defined on the predictable $\sigma$-field. We also explain how both cases are connected to recent developments in the theory of optimal portfolio choice, in particular to robust extensions of the classical expected utility criterion.

A general scheme of parameterized families of equations is considered, and abstract results on the expansion of the solutions and on the acceleration of their convergence in terms of the parameter are presented. These results are applied to fractional step approximations for linear parabolic PDEs, systems of linear PDEs, and for nonlinear ordinary differential equations. Applications to accelerating the convergence of finite difference schemes for these equations will be presented in a subsequent paper.

It is well known that a class of subordinators can be represented using the local time of Brownian motions. An extension of such a representation is given for a class of Lévy processes which are not necessarily of bounded variation. This class can be characterized by the complete monotonicity of the Lévy measures. The asymptotic behavior of such processes is also discussed and the results are applied to the generalized arc-sine law, an occupation time problem on the positive side for one-dimensional diffusion processes.

A complete analytic set is defined through the concepts of invariant measure mixing

We show that a certain model for the spread of an infection has a phase transition in the recuperation rate. The model is as follows: There are particles or individuals of type $A$ and type $B$, interpreted as healthy and infected, respectively. All particles perform independent, continuous time, simple random walks on $\mathbb{Z}^d$ with the same jump rate $D$. The only interaction between the particles is that at the moment when a $B$-particle jumps to a site which contains an $A$-particle, or vice versa, the $A$-particle turns into a $B$-particle. All $B$-particles recuperate (that is, turn back into $A$-particles) independently of each other at a rate $\la$. We assume that we start the system with $N_A(x,0-)$ $A$-particles at $x$, and that the $N_A(x,0-), \, x \in \mathbb{Z}^d$, are i.i.d., mean $\mu_A$ Poisson random variables. In addition we start with one additional $B$-particle at the origin. We show that there is a critical recuperation rate $\la_c > 0$ such that the $B$-particles survive (globally) with positive probability if $\la < \la_c$ and die out with probability 1 if $\la > \la_c$.

Let $\mu =\left({\mu }^1,\cdots ,{\mu }^d\right)$ be such that each ${\mu }^i$ is a signed measure on ${\mathrm{\backslash R}}^d$ belonging to the Kato class ${\mathrm{\backslash K}}_{d,1}$. The existence and uniqueness of a continuous Markov process $X$ on ${\mathrm{\backslash R}}^d$, called a Brownian motion with drift $\mu$, was recently established by Bass and Chen. In this paper we study the potential theory of $X$. We show that $X$ has a continuous density $q^{\mu }$ and that there exist positive constants $c_i$, $i=1,\cdots ,9$, such that $$c_1{e}^{-c_{2}t}{t}^{-\frac{d}{2}}{e}^{-\frac{c_3{∣ x-y∣ }^2}{2t}}\le q^{\mu }\left(t,x,y\right)\le c_4{e}^{c_{5}t}{t}^{-\frac{d}{2}}{e}^{-\frac{c_6{∣ x-y∣ }^2}{2t}}$$ and $$∣ {\nabla }_x{q}^{\mu }\left(t,x,y\right)∣ \le c_7{e}^{c_{8}t}{t}^{-\frac{d+1}{2}}{e}^{-\frac{c_9{∣ x-y∣ }^2}{2t}}$$ for all $\left(t,x,y\right)\in \left(0,\infty \right)\times {\mathrm{\backslash R}}^d\times {\mathrm{\backslash R}}^d$. We further show that, for any bounded $C^{1,1}$ domain $D$, the density $q^{\mu ,D}$ of $X^D$, the process obtained by killing $X$ upon exiting from $D$, has the following estimates: for any $T\&\mathrm{gt\; ;}0$, there exist positive constants $C_i$, $i=1,\cdots ,5,$ such that $$\begin{array}{c}{C}_1\left(1\wedge \frac{\rho \left(x\right)}{\sqrt{t}}\right)\left(1\wedge \frac{\rho \left(y\right)}{\sqrt{t}}\right)t^{-\frac{d}{2}}{e}^{-\frac{C_2{∣ x-y∣ }^2}{t}}\le q^{\mu ,D}\left(t,x,y\right)\\ \le C_3\left(1\wedge \frac{\rho \left(x\right)}{\sqrt{t}}\right)\left(1\wedge \frac{\rho \left(y\right)}{\sqrt{t}}\right)t^{-\frac{d}{2}}{e}^{-\frac{C_4{∣ x-y∣ }^2}{t}}\\ \end{array}$$ and $$∣ {\nabla }_x{q}^{\mu ,D}\left(t,x,y\right)∣ \le C_5\left(1\wedge \frac{\rho \left(y\right)}{\sqrt{t}}\right)t^{-\frac{d+1}{2}}{e}^{-\frac{C_4{∣ x-y∣ }^2}{t}}$$ for all $\left(t,x,y\right)\in \left(0,T\right]\times D\times D$, where $\rho \left(x\right)$ is the distance between $x$ and $\partial D$. Using the above estimates, we then prove the parabolic Harnack principle for $X$ and show that the boundary Harnack principle holds for the nonnegative harmonic functions of $X$. We also identify the Martin boundary of $X^D$.

Consider two independent sequences of travelers arriving at opposite ends of a one-lane shared pathway. Each traveler attempts to traverse the entire pathway to the opposite end. An attempt fails if the traveler collides with an opposing traveler. In a collision, both opposing travelers are annihilated. We study the probability that a traveler manages to traverse the entire length of the one-lane shared pathway unobstructed. The dynamics of the travelers include the possibility of acting as bodyguards and "running interference" for a more recent arrival traveling in the same direction. This model was developed to address some questions in the theory of crystal growth. It may have possible applications in Particle Physics as well as to traffic at a one-lane bridge. This paper develops some properties of the model while focusing on the probability that a traveler crosses the entire pathway without interference.

The Laplacian-$b$ random walk is a measure on self-avoiding paths that at each step has translation probabilities weighted by the $b$th power of the probability that a simple random walk avoids the path up to that point. We give a heuristic argument as to what the scaling limit should be and call this process the Laplacian-$b$ motion, $LM_b$. In simply connected domains, this process is the Schramm-Loewner evolution with parameter $\kappa = 6/(2b+1)$. In non-simply connected domains, it corresponds to the harmonic random Loewner chains as introduced by Zhan.

Given a measurable mapping $f$ from a nonatomic Loeb probability space $(T,\mathcal{T},P)$ to the space of Borel probability measures on a compact metric space $A$, we show the existence of a measurable mapping $g$ from $(T,\mathcal{T},P)$ to $A$ itself such that $f$ and $g$ yield the same values for the integrals associated with a countable class of functions on $T\times A$. A corollary generalizes the classical result of Dvoretzky-Wald-Wolfowitz on purification of measure-valued maps with respect to a finite target space; the generalization holds when the domain is a nonatomic, vector-valued Loeb measure space and the target is a complete, separable metric space. A counterexample shows that the generalized result fails even for simple cases when the restriction of Loeb measures is removed. As an application, we obtain a strong purification for every mixed strategy profile in finite-player games with compact action spaces and diffuse and conditionally independent information.

It has recently been proved that a continuous path of bounded variation in $\R^d$ can be characterised in terms of its transform into a sequence of iterated integrals called the signature of the path. The signature takes its values in an algebra and always has a logarithm. In this paper we study the radius of convergence of the series corresponding to this logarithmic signature for the path. This convergence can be interpreted in control theory (in particular, the series can be used for effective computation of time invariant vector fields whose exponentiation yields the same diffeomorphism as a time inhomogeneous flow) and can provide efficient numerical approximations to solutions of SDEs. We give a simple lower bound for the radius of convergence of this series in terms of the length of the path. However, the main result of the paper is that the radius of convergence of the full log signature is finite for two wide classes of paths (and we conjecture that this holds for all paths different from straight lines).

In the theory of progressive enlargements of filtrations, the supermartingale $Z_{t}=\mathbf{P}( g>t\mid \mathcal{F}_{t}) $ associated with an honest time $g$, and its additive (Doob-Meyer) decomposition, play an essential role. In this paper, we propose an alternative approach, using a multiplicative representation for the supermartingale $Z_{t}$, based on Doob's maximal identity. We thus give new examples of progressive enlargements. Moreover, we give, in our setting, a proof of the decomposition formula for martingales , using initial enlargement techniques, and use it to obtain some path decompositions given the maximum or minimum of some processes.

For a given element $f\in L^1$ and a convex cone $C\subset L^\infty$, $C\cap L^\infty_+=\{0\}$, we give necessary and sufficient conditions for the existence of an element $g\ge f$ lying in the polar of $C$. This polar is taken in $(L^\infty)^*$ and in $L^1$. In the context of mathematical finance the main result concerns the existence of martingale measures whose densities are bounded from below by a prescribed random variable.

Stochastic integrals of nonrandom $(l\times d)$-matrix-valued functions or nonrandom real-valued functions with respect to an additive process $X$ on $\mathbb{R}^d$ are studied. Here an additive process means a stochastic process with independent increments, stochastically continuous, starting at the origin, and having cadlag paths. A necessary and sufficient condition for local integrability of matrix-valued functions is given in terms of the Lévy--Khintchine triplets of a factoring of $X$. For real-valued functions explicit expressions of the condition are presented for all semistable Lévy processes on $\mathbb{R}^d$ and some selfsimilar additive processes. In the last part of the paper, existence conditions for improper stochastic integrals $\int_0^{\infty-}f(s)dX_s$ and their extensions are given; the cases where $f(s)\asymp s^{\beta} e^{-cs^{\alpha}}$ and where $f(s)$ is such that $s=\int_{f(s)}^{\infty} u^{-2} e^{-u} du$ are analyzed.

We consider a random perturbation of a pseudoperiodic flow on $\R^2$. The structure of such flows has been studied by Arnol'd; it contains regions where there are local Hamiltonians, and an ergodic region. Under an appropriate change of time, we identify a reduced model as the strength of the random perturbation tends to zero (along a certain subsequence). In the Hamiltonian region, arguments of Freidlin and Wentzell are used to identify a limiting graph-valued process. The ergodic region is reduced to a single point, which is "sticky". The identification of the glueing conditions which rigorously describe this stickiness follows from a perturbed test-function analysis in the ergodic region.

This essentially self-contained paper continues the study of a simple PDE in which an unorthodox sign in the spacial boundary condition destroys the usual Minimum Principle. The long-term behavior of solutions with time-parameter set $(0,\infty)$ is established, and this clarifies in analytic terms the characterization of non-negative solutions which had been obtained previously by probabilistic methods. The paper then studies by direct methods bounded 'ancient' solutions in which the time-parameter set is $(-\infty,0)$. In the final section, Martin-boundary theory is used to describe all non-negative ancient solutions in the most interesting case. The relevant Green kernel density behaves rather strangely, exhibiting two types of behavior in relation to scaling of its arguments. The Martin kernel density, a ratio of Green kernel densities, behaves more sensibly. Doob $h$-transforms illuminate the structure. As a somewhat surprising consequence of our Martin-boundary analysis, we find that non-negative solutions to our parabolic-looking equation satisfy an elliptic-type Harnack principle.

We show that there is a numerical scheme for the stochastic wave equation which converges in $L^p$ at a rate of $O(\sqrt h)$, and which converges a.s. uniformly on compact sets at a rate $O(\sqrt{ h|\log h|^\ep})$\,, for any $\ep >0$\,, where $h$ is the step size in both time and space. We show that this is the optimal rate: there is no scheme depending on the same increments of white noise which has a higher order of convergence.

We give a brief survey of Doob's work on cluster sets and fine limits.