A fixed two-dimensional projection of a three-dimensional Brownian motion is almost surely neighborhood recurrent; is this simultaneously true of all the two-dimensional projections with probability 1? Equivalently: three-dimensional Brownian motion hits any infinite cylinder with probability 1; does it hit all cylinders? This papers shows that the answer is no. Brownian motion in three dimensions avoids random cylinders and in fact avoids bodies of revolution that grow almost as fast as cones.

We consider the class of stationary, zero-mean Gaussian processes, indexed by the circle, satisfying a two-point Markov property and taking values in a vector bundle over the circle with given holonomy. We establish, subject to certain additional symmetry properties, a classification of all such processes. We then propose a construction of a Brownian motion of loops, in which these processes provide the infinitesimal increments.

A sequence of piecewise constant approximations to rescaled isotropic homeomorphic stochastic flows is shown to converge weakly in Skorohod metric to the coalescing Brownian flow. Intermittent behavior of isotropic flows is exposed, and the clustering properties of isotropic flows are studied by the means of this convergence. We obtain qualitative and quantitative description of expansions and contractions of an arbitrary isotropic homeomorphic flow on large time-and space-scales.

On prouve un résultat d’existence et d’unicité pour une généralisation (par introduction d’un opérateur maximal monotone multivoque) du problème de Skorohod (avec réflexion normale) déterministe associé àun convexe fermé $D$ de $\mathbb{R}^d$. La formulation Í partir des opérateurs maximaux monotones multivoques permet de considérer des drifts singuliers explosant au bord du domaine. Cette ‘‘approche multivoque’’ clarifie la connexion entre problème de Skorohod et semigroupes nonlinéaires. En guise d’application, on considère ensuite le cas stochastique: les équations différentielles stochastiques multivoques sont ainsi revisitées. Par conséquent, cette contribution fournit une nouvelle méthode de construction de diffusions réfléchies normalement possédant un coefficient de dérive discontinu, explosif.

An existence and uniqueness result is proven for a generalization (by introduction of a multivalued maximal monotone operator) of the deterministic Skorohod problem (with normal reflection) associated with a closed convex $D$ in $\mathbb{R}^d$ . The maximal monotone operator formulation allows for drifts that blow up as one gets near the boundary. This ‘‘multivalued approach’’ clarifies the connection between nonlinear semigroup theory and the Skorohod problem. As a consequence, we discuss then the stochastic case: multivalued stochastic differential equations are thus revisited. Therefore, we give an alternative way to construct diffusions with normal reflecting boundary conditions and discontinuous, exploding drift.

Fleming-Viot processes are probability-measure-valued diffusion processes that can be used as stochastic models in population genetics. Here we use duality methods to prove ergodic theorems for Fleming-Viot processes, including those with recombination. Coupling methods are also used to establish ergodicity of Fleming-Viot processes, first without and then with selection. A special type of selection known as symmetric overdominance is treated by other methods.

A general class of finite variance critical $(\xi, \Phi, k)$-superprocesses $X$ in a Luzin space $E$ with cadlag right Markov motion process $\xi$, regular local branching mechanism and branching functional $k$ of bounded characteristic are shown to continuously depend on $(\Phi, k)$. As an application we show that the processes with a classical branching functional $k(ds) = \varrho_s (\xi_s) ds [that is, a branching functional $k$ generated by a classical branching rate $\varrho_s (y)] are dense in the above class of $(\xi, \Phi, k)$-superprocesses $X$. Moreover, we show that, if the phase space $E$ is a compact metric space and $\xi$ is a Feller process, then always a Hunt version of the $(\xi, \Phi, k)$-superprocess $X$ exists. Moreover, under this assumption, we even get continuity in $(\Phi, k)$ in terms of weak convergence of laws on Skorohod path spaces.

We derive a large deviation principle for the occupation time func-tional, acting on functions with zero Lebesgue integral, for both super-Brownian motion and critical branching Brownian motion in three dimensions. Our technique, based on a moment formula of Dynkin, allows us to compute the exact rate functions, which differ for the two processes. Obtaining the exact rate function for the super-Brownian motion solves a conjecture of Lee and Remillard. We also show the corresponding CLT and obtain similar results for the superprocesses and critical branching process built over the symmetric stable process of index $\beta$ in $R^d$, with $d < 2\beta < 2 + d$ .

A conjecture of Liggett concerning the regime of weak survival for the contact process on a homogeneous tree is proved. The conjecture is shown to imply that the Hausdorff dimension of the limit set of such a contact process is no larger than half the Hausdorff dimension of the space of ends of the tree. The conjecture is also shown to imply that at the boundary between weak survival and strong survival, the contact process survives only weakly, a theorem previously proved by Zhang. Finally, a stronger form of a theorem of Hawkes and Lyons concerning the Hausdorff dimension of a Galton–Watson tree is proved.

The voter model, with mutations occurring at a positive rate $\alpha$, has a unique equilibrium distribution. We investigate the logarithms of the relative abundance of species for these distributions in $d \geq 2$. We show that, as $\alpha \to \infty$, the limiting distribution is right triangular in $d = 2$ and uniform in $d \geq 3$. We also obtain more detailed results for the histograms that biologists use to estimate the underlying density functions.

The excursion random measure $\zeta$ of a stationary process is defined on sets $E \subset (-\infty, \infty) \times (0, \infty)$, as the time which the process (suitably normalized) spends in the set E . Particular cases thus include a multitude of features (including sojourn times) related to high levels. It is therefore not surprising that a single limit theorem for $\zeta$ at high levels contains a wide variety of useful extremal and high level exceedance results for the stationary process itself.

The theory given for the excursion random measure demonstrates, under very general conditions, its asymptotic infinite divisibility with certain stability and independence of increments properties leading to its asymptotic distribution (Theorem 4.1). The results are illustrated by a number of examples including stable and Gaussian processes.

We derive upper and lower asymptotic bounds for the distribution of the supremum for a self-similar stochastic process. As an intermediate step, most proofs relate suprema to sojourns before proceeding to an appropriate discrete approximation.

Our results rely on one or more of three assumptions, which in turn essentially require weak convergence, existence of a first moment and tightness, respectively. When all three assumptions hold, the upper and lower bounds coincide (Corollary 1).

For P-smooth processes, weak convergence can be replaced with the use of a certain upcrossing intensity that works even for (a.s.) discontinuous processes (Theorem 7).

Results on extremes for a self-similar process do not on their own imply results for Lamperti’s associated stationary process or vice versa, but we show that if the associated process satisfies analogues of our three assumptions, then the assumptions hold for the self-similar process itself. Through this connection, new results on extremes for self-similar processes can be derived by invoking the stationarity literature.

Examples of application include Gaussian processes in $\mathbb{R}^n$, totally skewed $\alpha$-stable processes, Kesten–Spitzer processes and Rosenblatt processes.

We consider the problem of $L_p$-consistent density estimation from the initial segments of strongly dependent processes. It is shown that no procedure can consistently estimate the one-dimensional marginal density of every stationary ergodic process for which such a density exists. A similar result is established for the problem of estimating the support of the marginal distribution of an ergodic process.

We extend Lévy’s classical criterion for a sequence of independent identically distributed random variables to belong to the domain of partial attraction of a nondegenerate Gaussian law to stationary $\phi$-mixing sequences. We also extend some results of Kesten and of Kuelbs and Zinn on the LIL behavior of independent identically distributed random variables to stationary $\phi$-mixing sequences. No assumptions on the rate of decay for the mixing coefficient are made.

Let $\xi, \xi_1, \dots$ be i.i.d. real-valued random variables and $S_n = \xi_1 + \dots + \xi_n$. In the case when the distribution of $\xi$ is close to a stable $(\alpha)$ law for some $\alpha \epsilon (0, 1) \bigcup (1, 2)$, we investigate the asymptotic behavior in distribution of the maximum of normalized sums, $\max_{k=1,\dots,n} k^{-1/\alpha}S_k$. This completes the Darling-Erdös limit theorem for the case $\alpha = 2$.

After introducing a new concept, the notion of $\mathscr{E}$-martingale, we extend the well-known Doob inequality (for $1 < p < +\infty)$ and the Burkholder–Davis–Gundy inequalities (for $p = 2$) to $\mathscr{E}$-martingales. By means of these inequalities, we give sufficient conditions for the closedness of a space of stochastic integrals with respect to a fixed $\mathbb{R}^d$-valued semimartingale, a question which arises naturally in the applications to financial mathematics. We also provide a necessary and sufficient condition for the existence and uniqueness of the Föllmer–Schweizer decomposition.

In this paper, we study the weak convergence to an appropriate Gaussian process of the empirical process of the block-based bootstrap estimator proposed by Künsch for stationary sequences. The classes of processes investigated are weak dependent and associated sequences. We also prove that, differently from the independent situation, the bootstrapped estimator of the mean of certain dependent sequences satisfies the central limit theorem while the mean of the original sequence does not.

Let $m \geq 2$ be a nonnegative integer and let ${X^{(l)}, X_i^{(l)}}_{i \epsilon \mathbb{N}}, l = 1, \dots, m$, be $m$ independent sequences of independent and identically distributed symmetric random variables. Define $S_n = \Sigma_{1 \leq i_1, \dots, i_m \leq n} X_{i_1}^{(l)} \dots X_{i_m}^{(m)}$, and let ${\gamma_n}_{n \epsilon \mathbb{N}}$ be a nondecreasing sequence of positive numbers, tending to infinity and satisfying some regularity conditions. For $m = 2$ necessary and sufficient conditions are obtained for the strong law of large numbers $\gamma_n^{-1} S_n \to 0$ a.s. to hold, and for $m > 2$ the strong law of large numbers is obtained under a condition on the growth of the truncated variance of the $X^{(l)}$ .