Let $\{X_t\}_{t \geq 0}$ be a symmetric, nearest-neighbor random walk on $\mathbb{Z}^d$ with exponential holding times of expectation $1/d$, starting at the origin. For a potential $V: \mathbb{Z}^d \to [0, \infty)$ with finite and nonempty support, define transformed path measures by $d \hat{\mathbb{P}}_T \equiv \exp (T^{-1} \int_0^T \int_0^T V(X_s - X_t) ds dt) d \mathbb{P}/Z_T$ for $T > 0$, where $Z_T$ is the normalizing constant. If $d = 1$ or if the self-attraction is sufficiently strong, then $||X_t||_{\infty}$ has an exponential moment under $\hat{\mathbb{P}}_T$ which is uniformly bounded for $T > 0$ and $t \in [0, T]$. We also prove that ${X_t}_{t \geq 0}$ under suitable subsequences of $\{\hat{\mathbb{P}}_T\}_{T > 0}$ behaves for large $T$ asymptotically like a mixture of space-inhomogeneous ergodic random walks. For special cases like a sufficiently strong Dirac-type interaction, we even prove convergence of the transformed path measures and the law of $X_T$ as well as of the law of the empirical measure $L_T$ under $\{\hat{\mathbb{P}}_T\}_{T > 0}$.

The Edwards model in one dimension is a transformed path measure for standard Brownian motion discouraging self-intersections. We prove a central limit theorem for the endpoint of the path, extending a law of large numbers proved by Westwater. The scaled variance is characterized in terms of the largest eigenvalue of a one-parameter family of differential operators, introduced and analyzed by van der Hofstad and den Hollander. Interestingly, the scaled variance turns out to be independent of the strength of self-repellence and to be strictly smaller than one (the value for free Brownian motion).

We study the fluctuation problem for the multiple point range of random walks in the two dimensional integer lattice with mean 0 and finite variance. The $p$-multiple point range means the number of distinct sites with multiplicity $p$ of random walk paths before time $n$. The suitably normalized multiple point range is proved to converge to a constant, which is independent of the multiplicity, multiple of the renormalized self-intersection local time of a planar Brownian motion.

We investigate natural linear additive (NLA) functionals of a general critical $(\xi, K, \psi)$-superprocess $X$. We prove that all of them have only fixed discontinuities. All homogeneous NLA functionals of time-homogeneous superprocesses are continuous (this was known before only in the case of quadratic branching).

We introduce an operator $\mathscr{E}(u)$ defined in terms of $(\xi, K, \psi)$ and we prove that the potential $h$ and the log-potential $u$ of a NLA functional $A$ are connected by the equation $u + \mathscr{E}(u) = h$. The potential is always an exit rule for $\xi$ and the condition $h + \mathscr{E}(h) < \infty$ a.e. is sufficient for an exit rule $h$ to be a potential.

In an accompanying paper, these results are applied to boundary value problems for partial differential equations involving nonlinear operator $Lu = u^{\alpha}$ where $L$ is a second order elliptic differential operator and $1 < \alpha \leq 2$.

Suppose that $E$ is an arbitrary domain in $\mathbb{R}^d$, $L$ is a second order elliptic differential operator in $S = \mathbb{R}_+ \times E$ and $S^e$ is the extremal part of the Martin boundary for the corresponding diffusion $\xi$. Let $1 < \alpha \leq 2$. We investigate a boundary value problem

$$\tag{*}\begin{eqnarray} \frac{\partial u}{\partial r}+Lu−u^\alpha&=&−\eta \quad \textrm{in }S,\\ u&=&v \quad \textrm{on }S^e,\\ u&=&0 \quad \textrm{on }\{\infty\} × E \end{eqnarray}$$

involving two measures $\eta$ and $\nu$. For the existence of a solution, we give sufficient conditions in terms of a Martin capacity and necessary conditions in terms of hitting probabilities for an $(L, \alpha)$-superdiffusion $X$. If a solution exists, then it can be expressed by an explicit formula through an additive functional $A$ of $X$.

An $(L, \alpha)$-superdiffusion is a branching measure-valued process. A natural linear additive (NLA) functional $A$ of $X$ is determined uniquely by its potential $h$ defined by the formula $P_{\mu} A(0, \infty) = \int h(r, x) \mu (dr, dx)$ for all $\mu \in \mathscr{M}^*$ (the determining set of $A$). Every potential $h$ is an exit rule for $\xi$ and it has a unique decomposition into extremal exit rules. If $\eta$ and $\nu$ are measures which appear in this decomposition, then (*)can be replaced by an integral equation

$$\tag{**} u(r, x) + \int p(r, x; t, dy)u(t, y)^{\alpha} ds = h(r, x),$$

where $p(r, x; t, dy)$ is the transition function of $\xi$. We prove that h is the potential of a NLA functional if and only if (**) has a solution $u$. Moreover,

$$u(r, x) = -\log P_{r, x} e^{-A(0, \infty)}.$$

By applying these results to homogeneous functionals of time-homogeneous superdiffusions, we get a stronger version of theorems proved in an earlier publication. The foundation for our present investigation is laid by a general theory developed in the accompanying paper.

We study reflected solutions of one-dimensional backward stochastic differential equations. The “reflection” keeps the solution above a given stochastic process. We prove uniqueness and existence both by a fixed point argument and by approximation via penalization. We show that when the coefficient has a special form, then the solution of our problem is the value function of a mixed optimal stopping–optimal stochastic control problem. We finally show that, when put in a Markovian framework, the solution of our reflected BSDE provides a probabilistic formula for the unique viscosity solution of an obstacle problem for a parabolic partial differential equation.

Soit $M$ un espace métrique séparable complet àcourbure négative suivant la définition de Herer. À l’aide de la définition de Herer de l’espérance mathématique d’un point aléatoire de $M$, nous étendons àdes suites de points aléatoires de $M$ un théorème ergodique ponctuel et plusieurs lois fortes des grands nombres (lFgn), connus dans le cas où $M$ est un espace de Banach séparable (lFgn d’Etemadi, de Beck et Giesy, et de Cuesta et Matrán). Dans les résultats obtenus, la convergence a lieu au sens de Hausdorff ou au sens de Wijsman dans l’espace des fermés de $M$.

Let $M$ be a complete separable metric space with negative curvature as defined by Herer. Using Herer’s definition of the mathematical expectation of a random point of $M$, we extend to sequences of random points of $M$ a pointwise ergodic theorem and strong laws of large numbers (SLLN), known in the case where $M$ is a separable Banach space (SLLN of Etemadi, of Beck and Giesy and of Cuesta and Matrán). The convergence results obtained here are stated for the Hausdorff topology or the Wijsman topology in the space of closed subsets of $M$.

The well-known Cramér-Wold theorem states that a Borel probability measure on $\mathbb{R}^d$ is uniquely determined by the totality of its one-dimensional projections. In this paper we examine various conditions under which a probability measure is determined by a subset of its $(d - 1)$-dimensional orthogonal projections.

Let $\{S_n: n \geq 0\}$ be a random walk having normally distributed increments with mean $\theta$ and variance 1, and let $\tau$ be the time at which the random walk first takes a positive value, so that $S_{\tau}$ is the first ladder height. Then the expected value $E_{\theta} S_{\tau}$, originally defined for positive $\theta$, maybe extended to be an analytic function of the complex variable $\theta$ throughout the entire complex plane, with the exception of certain branch point sin-gularities. In particular, the coefficients in a Taylor expansion about $\theta = 0$ may be written explicitly as simple expressions involving the Riemann zeta function. Previously only the first coefficient of the series developed here was known; this term has been used extensively in developing approximations for boundary crossing problems for Gaussian random walks. Knowledge of the complete series makes more refined results possible; we apply it to derive asymptotics for boundary crossing probabilities and the limiting expected overshoot.

Let $\Phi (\cdot)$ be a nondecreasing convex function on $[0, \infty)$. We show that for any integer $n \geq 1$ and real $a$, $$E \Phi ((M_n - a)^+) \leq 2E \Phi ((S_n - a)^+) - \Phi (0)$$ and $$E(M_n \vee \mathrm{med} S_n) \leq E|S_n - \mathrm{med} S_n|.$$ where $X_1, X_2, \dots$ are any independent mean zero random variables with partial sums $S_0 = 0, S_k = X_1 + \dots + X_k$ and partial sum maxima $M_n = \max_{0 \leq k \leq n} S_k$. There are various instances in which these inequalities are best possible for fixed $n$ and/or as $n \to \infty$. These inequalities remain valid if $\{X_k\}$ is a martingale difference sequence such that $E(X_k \{X_i; i \not= k\}) = 0$ a.s. for each $k \geq 1$. Modified versions of these inequalities hold if the variates have arbitrary means but are independent.

Let $(B^t (s), 0 \leq s < \infty)$ be reflecting inhomogeneous Brownian motion with drift $t - s$ at time $s$, started with $B^t (0) = 0$. Consider the random graph $\mathscr{G}(n, n^{-1} + tn^{-4/3})$, whose largest components have size of order $n^{2/3}$. Normalizing by $n^{-2/3}$, the asymptotic joint distribution of component sizes is the same as the joint distribution of excursion lengths of $B^t$ (Corollary 2). The dynamics of merging of components as $t$ increases are abstracted to define the multiplicative coalescent process. The states of this process are vectors $\mathsf{x}$ of nonnegative real cluster sizes $(x_i)$, and clusters with sizes $x_i$ and $x_j$ merge at rate $x_i x_j$. The multiplicative coalescent is shown to be a Feller process on $l_2$. The random graph limit specifies the standard multiplicative coalescent, which starts from infinitesimally small clusters at time $-\infty$; the existence of such a process is not obvious.

The two-parameter Poisson-Dirichlet distribution, denoted $\mathsf{PD}(\alpha, \theta)$ is a probability distribution on the set of decreasing positive sequences with sum 1. The usual Poisson-Dirichlet distribution with a single parameter $\theta$, introduced by Kingman, is $\mathsf{PD}(0, \theta)$. Known properties of $\mathsf{PD}(0, \theta)$, including the Markov chain description due to Vershik, Shmidt and Ignatov, are generalized to the two-parameter case. The size-biased random permutation of $\mathsf{PD}(\alpha, \theta)$ is a simple residual allocation model proposed by Engen in the context of species diversity, and rediscovered by Perman and the authors in the study of excursions of Brownian motion and Bessel processes. For $0 < \alpha < 1, \mathsf{PD}(\alpha, 0)$ is the asymptotic distribution of ranked lengths of excursions of a Markov chain away from a state whose recurrence time distribution is in the domain of attraction of a stable law of index $\alpha$. Formulae in this case trace back to work of Darling, Lamperti and Wendel in the 1950s and 1960s. The distribution of ranked lengths of excursions of a one-dimensional Brownian motion is $\mathsf{PD}(1/2, 0)$, and the corresponding distribution for a Brownian bredge is $\mathsf{PD}(1/2, 1/2)$. The $\mathsf{PD}(\alpha, 0)$ and $\mathsf{PD}(\alpha, \alpha)$ distributions admit a similar interpretation in terms of the ranked lengths of excursions of a semistable Markov process whose zero set is the range of a stable subordinator of index $\alpha$.

Schrödinger processes are defined as mixtures of Brownian bridges which preserve the Markov property. In finite dimensions, they can be characterized as $h$-transforms in the sense of Doob for some space-time harmonic function $h$ of Brownian motion, and also as solutions to a large deviation problem introduced by Schrödinger which involves minimization of relative entropy with given marginals. As a basic case study in infinite dimensions, we investigate these different aspects for Schrödinger processes of infinite-dimensional Brownian motion. The results and examples concerning entropy minimization with given marginals are of independent interest.

We derive inequalities of the form $\Delta (P, Q) \leq H(P|R) + H(Q|R)$ which hold for every choice of probability measures P, Q, R, where $H(P|R)$ denotes the relative entropy of $P$ with respect to $R$ and $\Delta (P, Q)$ stands for a coupling type "distance" between $P$ and $Q$. Using the chain rule for relative entropies and then specializing to $Q$ with a given support we recover some of Talagrand's concentration of measure inequalities for product spaces.

In this paper we study the asymptotic behavior of additive functionals of Brownian motion which are not necessarily of bounded variation. The result is then applied to the Hilbert transform of the Brownian local time.

Soit $(B_t)_{t \in \mathbb{R}_+}$ un mouvement brownien dans $\mathbb{R}$, issu de 0. Soit $(L_t^x)_{t \in \mathbb{R}_+}^{x \in \mathbb{R}}$ une version continue de ses temps locaux. $F$ étant un fermé de $\mathbb{R}$, contenant 0, on s'intéresse au processus càdlàg $(A_t^F)_{t \in \mathbb{R}_+}$, où $A_t^F$ est "le" point de $F$ le plus visité àl'instant $t$, c'est-à-dire "le" point $a \in F$ tel que $L_t^a = L_t^F$, en notant $L_t^F = \max_{x \in F} L_t^x$.

On démontre que le processus $(A_t^F)_{t \in \mathbb{R}_+}$ est àvariation localement bornée (et même purement de sauts) et que le processus croissant $(L_t^F)_{t \in \mathbb{R}_+}$ majore le temps local symétrique en 0 de la semi-martingale $(B_t - A_t^F)_{t \in \mathbb{R}_+}$.

Lorsque $F = \mathbb{R}$, on montre que cette majoration est en fait une égalité et que les sauts du processus $(A_t^F)_{t \in \mathbb{R}_+}$ ne sont ni isolés, ni "obligés."

We provide an integral representation for smooth functionals of continuous semimartingales. The representation is related to an infinite-dimensional nonautonomous parabolic equation. Semimartingale integral representations, including a martingale representation, are given in terms of a solution to this equation.