Constrained diffusions in convex polyhedral cones with a general oblique reflection field, and with a diffusion coefficient scaled by a small parameter $\varepsilon> 0$, are considered. Using an interior Dirichlet heat kernel lower bound estimate for second order elliptic operators in bounded domains from Zhang (1995), certain uniform in $\varepsilon$ lower bounds on transition densities of such constrained diffusions are established. These lower bounds together with results from Biswas & Budhiraja (2011) give, under additional stability conditions, an exponential leveling property as $\varepsilon \to 0$ for exit times from suitable bounded domains.

In this paper, we study the large deviations for Engel's series, Sylvester's series and Cantor's products from number theory.

Random walks in random sceneries (RWRS) are simple examples of stochastic processes in disordered media. They were introduced at the end of the 70's by Kesten-Spitzer and Borodin, motivated by the construction of new self-similar processes with stationary increments. Two sources of randomness enter in their definition: a random field $\xi = (\xi(x))_{x \in \mathbb{Z}^d}$ of i.i.d. random variables, which is called the random scenery, and a random walk $S = (S_n)_{n \in \mathbb{N}}$ evolving in $\mathbb{Z}^d$, independent of the scenery. The RWRS $Z = (Z_n)_{n \in \mathbb{N}}$ is then defined as the accumulated scenery along the trajectory of the random walk, i.e., $Z_n := \sum_{k=1}^n \xi(S_k)$. The law of $Z$ under the joint law of $\xi$ and $S$ is called "annealed'', and the conditional law given $\xi$ is called "quenched''. Recently, functional central limit theorems under the quenched law were proved for $Z$ by the first two authors for a class of transient random walks including walks with finite variance in dimension $d \ge 3$. In this paper we extend their results to dimension $d=2$.

In this text, we consider an $N$ by $N$ random matrix $X$ such that all but $o(N)$ rows of $X$ have $W$ non identically zero entries, the other rows having less than $W$ entries (such as, for example, standard or cyclic band matrices). We always suppose that $1 \ll W \ll N$. We first prove that if the entries are independent, centered, have variance one, satisfy a certain tail upper-bound condition and $W \gg (\log N)^{6(1+\alpha)}$, where $\alpha$ is a positive parameter depending on the distribution of the entries, then the largest eigenvalue of $X/\sqrt{W}$ converges to the upper bound of its limit spectral distribution, that is $2$, as for Wigner matrices. This extends some previous results by Khorunzhiy and Sodin where less hypotheses were made on $W$, but more hypotheses were made about the law of the entries and the structure of the matrix. Then, under the same hypotheses, we prove a delocalization result for the eigenvectors of $X$, precisely that most of them cannot be essentially localized on less than $W/\log(N)$ entries. This lower bound on the localization length has to be compared to the recent result by Steinerberger, which states that the localization length in the edge is $\ll W^{7/5}$ or there is strong interaction between two eigenvectors in an interval oflength $W^{7/5}$.</p>

The asymptotic behavior of the tail probabilities for the first hitting times of the Bessel process with arbitrary index is shown without using the explicit expressions for the distribution function obtained in the authors' previous works.

We study protected nodes in various classes of random rooted trees by putting them in the general context of fringe subtrees introduced by Aldous (1991). Several types of random trees are considered: simply generated trees (or conditioned Galton-Watson trees), which includes several cases treated separately by other authors, binary search trees and random recursive trees. This gives unified and simple proofs of several earlier results, as well as new results.

An expansion of the Weyl function of a $H$-selfadjoint random matrix with one negative square is provided. It is shown that the coefficients converge to a certain generalization of Catlan numbers. Properties of this generalization are studied, in particular, a combinatorial interpretation is given.

Distributional findings are obtained relative to various quantities arising in Bernoulli arrays $\{ X_{k,j}, k \geq 1, j =1, \ldots, r+1\}$, where the rows $(X_{k,1}, \ldots, X_{k,r+1})$ are independently distributed as $\hbox{Multinomial}\,.$

We model two-dimensional crystals by a configuration space in which every admissible configuration is a hard disk configuration and a perturbed version of some triangular lattice with side length one. In this model we show that, under the uniform distribution, expected configurations in a given box are arbitrarily close to some triangular lattice whenever the particle density is chosen sufficiently high. This choice can be made independent of the box size.

In this paper, consider the circular Cauchy distribution $\mu_x$ on the unit circle $S$ with index $0\le |x|<1$, we study the spectral gap and the optimal logarithmic Sobolev constant for $\mu_x$, denoted respectively as $\lambda_1(\mu_x)$ and $C_{\mathrm{LS}}(\mu_x).$ We prove that $\frac{1}{1+|x|}\le \lambda_1(\mu_x)\le 1$ while $C_{\mathrm{LS}}(\mu_x)$ behaves like $\log(1+\frac{1}{1-|x|})$ as $|x|\to 1.$

A multi-class M/M/1 system, with service rate $\mu_in$ for class-$i$ customers, is considered with the risk-sensitive cost criterion $n^{-1}\log E\exp\sum_ic_iX^n_i(T)$, where $c_i>0$, $T>0$ are constants, and $X^n_i(t)$ denotes the class-$i$ queue-length at time $t$, assuming the system starts empty. An asymptotic upper bound (as $n\to\infty$) on the performance under a fixed priority policy is attained, implying that the policy is asymptotically optimal when $c_i$ are sufficiently large. The analysis is based on the study of an underlying differential game.

When an i.i.d. sequence of letters is cut into words according to i.i.d. renewal times, an i.i.d.\ sequence of words is obtained. In the annealed large deviation principle (LDP) for the empirical process of words, the rate function is the specific relative entropy of the observed law of words w.r.t. the reference law of words. In Birkner, Greven and den Hollander, the quenched LDP (= conditional on a typical letter sequence) was derived for the case where the renewal times have an algebraic tail. The rate function turned out to be a sum of two terms, one being the annealed rate function, the other being proportional to the specific relative entropy of the observed law of letters w.r.t. the reference law of letters, obtained by concatenating the words and randomising the location of the origin. The proportionality constant equals the tail exponent of the renewal process. The purpose of the present paper is to extend both LDP's to letter sequences that are not i.i.d. It is shown that both LDP's carry over when the letter sequence satisfies a mixing condition called summable variation. The rate functions are again given by specific relative entropies w.r.t. the reference law of words, respectively, letters. But since neither of these reference laws is i.i.d., several approximation arguments are needed to obtain the extension.

We are interested in the differential equations satisfied by the density of the Geometric Stable processes $\mathcal{G}_{\alpha }^{\beta }=\left\{\mathcal{G}_{\alpha }^{\beta }(t);t\geq 0\right\} $, with stability \ index $\alpha \in (0,2]$ and symmetry parameter $\beta \in \lbrack -1,1]$, both in the univariate and in the multivariate cases. We resort to their representation as compositions of stable processes with an independent Gamma subordinator. As a preliminary result, we prove that the latter is governed by a differential equation expressed by means of the shift operator. As a consequence, we obtain the space-fractional equation satisfied by the density of $\mathcal{G}_{\alpha }^{\beta }$. For some particular values of $\alpha $ and $\beta $, we get some interesting results linked to well-known processes, such as the Variance Gamma process and the first passage time of the Brownian motion.

We prove that any vertex-reinforced random walk on the integer lattice with non-decreasing reinforcement sequence $w$ satisfying $w(k) = o(k^{\alpha})$ for some $\alpha <1/2$ is recurrent. This improves on previous results of Volkov (2006) and Schapira (2012).

This is an Erratum for ECP volume 18 paper 72 (2013). The proof of statement 2) in Theorem 1.1 may not hold true. Indeed it relies on Lemma 2.5 which claims the transience of a certain Markov chain. The proof of Lemma 2.5 is incomplete because the arguments presented. In this note, we establish this transience.

We answer a question by Kolb and Wubker on the threshold drift for Brownian motion on an interval with redistribution from the boundary. We do that by constructing an efficient coupling.

We prove that the noisy voter model mixes extremely fast - in time of O(log(n)) on any graph with n vertices - for arbitrarily small values of the "noise parameter". We then explain why, as a result, this is an example of a spin system that is always in the "high-temperature regime".

We study the site version of (independent) first-passage percolation on the triangular lattice $T$. Denote the passage time of the site $v$ in $T$ by $t(v)$, and assume that $\mathbb{P}(t(v)=0)=\mathbb{P}(t(v)=1)=1/2$. Denote by $a_{0,n}$ the passage time from 0 to (n,0), and by b_{0,n} the passage time from 0 to the halfplane $\{(x,y) : x\geq n\}$. We prove that there exists a constant $0<\mu<\infty$ such that as $n\rightarrow\infty$, $a_{0,n}/\log n\rightarrow \mu$ in probability and $b_{0,n}/\log n\rightarrow \mu/2$ almost surely. This result confirms a prediction of Kesten and Zhang. The proof relies on the existence of the full scaling limit of critical site percolation on $T$, established by Camia and Newman.

We introduce a setup of model uncertaintyin discrete time. In this setup wederive dual expressions for the super-replication prices of game options with upper semicontinuous payoffs. We show that the super-replication price is equal to the supremum over a special (non dominated) set of martingale measures, of the corresponding Dynkin games values. This type of results is also new for American options.

This is an erratum for ECP volume 18 paper 65 (2013). In Proposition 3.1, Condition (C) does not imply that the set Λ(Γ) generates a dense subgroup of R. This has to be made an assumption. Alternatively, one can assume that the matrices are invertible.

We consider the standard site percolation model on the three dimensional cubic lattice. Starting solely with the hypothesis that $\theta(p)>0$, we prove that, for any $\alpha>0$, there exists $\kappa>0$ such that, with probability larger than $1-1/n^\alpha$, every pair of sites inside the box $\Lambda(n)$ are joined by a path having at most $\kappa(\ln n)^2$ closed sites.

We show that any càdlàg predictable process of finite variation is an a.s. limit of elementary predictable processes; it follows that predictable stopping times can be approximated "from below" by predictable stopping times which take finitely many values. We then obtain as corollaries two classical theorems: predictable stopping times are announceable, and an increasing process is predictable iff it is natural.

A 1-2 model configuration is a subset of edges of the hexagonal lattice such that each vertex is incident to one or two edges. We prove that for any translation-invariant Gibbs measure of 1-2 model, almost surely the infinite homogeneous cluster is unique.

We consider controlled random walks that are martingales with uniformly bounded increments and nontrivial jump probabilities and show that such walks can be constructed so that $P(S_n^u=0)$ decays at polynomial rate $n^{-\alpha}$ where $\alpha>0$ can be arbitrarily small. We also show, by means of a general delocalization lemma for martingales, which is of independent interest, that slower than polynomial decay is not possible.

Assume that $X$ is a real-valued martingale starting from $0$, $H$ is a predictable process with values in $[-1,1]$ and $Y$ is the stochastic integral of $H$ with respect to $X$. The paper contains the proofs of the following sharp weak-type estimates. (i) If $X$ has continuous paths, then $$ \mathbb{P}\left(\sup_{t\geq 0}|Y_t|\geq 1\right)\leq 2\mathbb{E} \sup_{t\geq 0}X_t.$$<br />(ii) If $X$ is arbitrary, then$$ \mathbb{P}\left(\sup_{t\geq 0}|Y_t|\geq 1\right)\leq 3.477977\ldots\mathbb{E} \sup_{t\geq 0}X_t.$$The proofs rest on Burkholder's method and exploits the existence of certain special functions possessing appropriate concavity and majorization properties.

We give an estimate for the Kolmogorov distance between an infinitely divisible distribution (with mean zero and variance one) and the standard Gaussian distribution in terms of the difference between the fourth moment and 3. In a similar fashion we give an estimate for the Kolmogorov distance between a freely infinitely divisible distribution and the Semicircle distribution in terms of the difference between the fourth moment and 2.

Consider a walker performing a random walk in an i.i.d. random environment, and assume that the walker tells us at each time the environment it sees at its present location. Given this history of the transition probabilities seen from the walker - but not its trajectory - can we reconstruct the law of the environment? We show that in a one-dimensional environment, the law of the environment can be reconstructed. This model can be seen as a special case of a scenery reconstruction problem, where the steps of the random walker depend on the scenery.<br />

We demonstrate a novel strong law of large numbers for branching processes, with a simple proof via measure-theoretic manipulations and spine theory. Roughly speaking, any sequence of events that eventually occurs almost surely for the spine entails the almost sure convergence of a certain sum over particles in the population.

We study stochastic perturbed Volterra equations of convolution type in an infinite dimensional case. Our interest is directed towards the existence and regularity of stochastic convolutions connected to the equations considered under some kind of perturbations. We use an operator theoretical method for the representation of solutions.

In a recent work, Blanc, Le Bris, and Lions defined a notion of increment-stationarity for random point sets, which allowed them to prove the existence of a thermodynamic limit for two-body potential energies on such point sets (under the additional assumption of ergodicity), and to introduce a variant of stochastic homogenization for increment-stationary coefficients. Whereas stationary random point sets are increment-stationary, it is not clear a priori under which conditions increment-stationary random point sets are stationary.In the present contribution, we give a characterization of the equivalence of both notions of stationarity based on elementary PDE theory in the probability space.This allows us to give conditions on the decay of a covariance function associated with the random point set, which ensure that increment-stationary random point sets are stationary random point sets up to a random translation with bounded second moment in dimensions $d>2$. In dimensions $d=1$ and $d=2$, we show that such sufficient conditions cannot exist.

Following Bertoin who considered the ergodicity and exponential decay of Lévy processes in a finite domain, we consider general Lévy processes and their ergodicity and exponential decay in a finite interval. More precisely, given $T_a=\inf\{t>0:\,X_t\notin $. Under general conditions, e.g. absolute continuity of the transition semigroup of the unkilled Lévy process, we prove that the killed semigroup is a compact operator. Thus, we prove stronger results in view of the exponential ergodicity and estimates of the speed of convergence. Our results are presented in a Lévy processes setting but are well applicable for Markov processes in a finite interval under information about Lebesgue irreducibility of the killed semigroup and that the killed process is a double Feller process. For example, this scheme is applicable to a work of Pistorius.<br />

Let $M_n$ denote the number of sites in the largest cluster in site percolation on the triangular lattice inside a box side length $n$. We give lower and upper bounds on the probability that $M_n / \mathbb{E} M_n > x$ of the form $\exp(-Cx^{2/\alpha_1})$ for $x \geq 1$ and large $n$ with $\alpha_1 = 5/48$ and $C>0$. Our results extend to other two dimensional lattices and strengthen the previously known exponential upper bound derived by Borgs, Chayes, Kesten and Spencer [BCKS99]. Furthermore, under some general assumptions similar to those in [BCKS99], we derive a similar upper bound in dimensions $d > 2$.

We extend the proof of the local semicircle law for generalized Wigner matrices given in [4] to the case when the matrix of variances has an eigenvalue $-1$. In particular, this result provides a short proof of the optimal local Marchenko-Pastur law at the hard edge (i.e. around zero) for sample covariance matrices $\boldsymbol{\mathrm{X}}^\ast \boldsymbol{\mathrm{X}} $, where the variances of the entries of $ \boldsymbol{\mathrm{X}} $ may vary.

We show that there is a one-to-one correspondence between diffusions and the solutions of the Skorokhod Embedding Problem due to Bertoin and Le-Jan. In particular, the minimal embedding corresponds to a "minimal local martingale diffusion", which is a notion we introduce in this article. Minimality is closely related to the martingale property. A diffusion is minimal if it minimises the expected local time at every point among all diffusions with a given distribution at an exponential time. Our approach makes explicit the connection between the boundary behaviour, the martingale property and the local time characteristics of time-homogeneous diffusions.

Let $v:[0,T]\times {\mathbf R}^d \to {\mathbf R}$ be the solution of the parabolic backward equation $$\partial_t v + (1/2) \sum_{i,j} [\sigma \sigma^\top]_{i,j} \partial_{x_i}\partial_{x_j}v+ \sum_{i} b_i \partial_{x_i}v + kv =0$$ with terminal condition $g$, where the coefficients are time-and state-dependent, and satisfy certain regularity assumptions. Let $X = (X_t)_{t\in [0,T]}$ be the associated ${\mathbf R}^d$-valued diffusion process on some appropriate $(\Omega,{\mathcal F},{\mathbb Q})$. For $p\in [2,\infty)$ and a measure $d{\mathbb P}=\lambda_T d{\mathbb Q}$, where $\lambda_T$ satisfies the Muckenhoupt condition $A_p$, we relate the behavior of \[ \|g(X_T)-{\mathbf E}_{\mathbb P}(g(X_T)|{\mathcal F}_t) \|_{L_p({\mathbb P})}, \quad \|\nabla v(t,X_t) \|_{L_p({\mathbb P})}, \quad \|D^2 v(t,X_t) \|_{L_p({\mathbb P})} \]to each other, where $D^2v:=(\partial_{x_i}\partial_{x_j}v)_{i,j}$ is the Hessian matrix.

We study the optimization of the joint $(p^Y,p^Z)$-variations of two continuous semimartingales $(Y,Z)$ driven by the same Itô process $X$. The $p$-variations are defined on random grids made of finitely many stopping times. We establish an explicit asymptotic lower bound for our criterion, valid in rather great generality on the grids, and we exhibit minimizing sequences of hitting time form. The asymptotics is such that the spatial increments of $X$ and the number of grid points are suitably converging to 0 and $+\infty$ respectively.

In this paper, the large deviations at the trajectory level for ergodic Markov processes are studied. These processes take values in the non-negative quadrant of the two-dimensional lattice and are concentrated on step-wise functions. The rates of jumps towards the axes (downward jumps) depend on the position of the process - the higher the position, the greater the rate. The rates of jumps going in the same direction as the axes (upward jumps) are constants. Therefore the processes are ergodic. The large deviations are studied under equal scalings of both space and time. The scaled versions of the processes converge to 0. The main result is that the probabilities of excursions far from 0 tend to 0 exponentially fast with an exponent proportional to the square of the scaling parameter. The proportionality coefficient is an integral of a linear combination of path components. A rate function of the large deviation principle is calculated for continuous functions only.

This paper examines an optimal investment problem in a continuous-time (essentially) complete financial market with a finite horizon. We deal with an investor who behaves consistently with principles of Cumulative Prospect Theory, and whose utility function on gains is bounded above. The well-posedness of the optimisation problemis trivial, and a necessary condition for the existence of an optimal trading strategyis derived. This condition requires that the investor’s probability distortion function on losses does not tend to 0 near 0 faster than a given rate, which is determined by the utility function. Under additional assumptions, we show that this condition is indeed the borderline for attainability, in the sense that for slower convergence of the distortion function there does exist an optimal portfolio.

In discrete contexts such as the degree distribution for a graph, scale-free has traditionally been defined to be power-law. We propose a reasonable interpretation of scale-free, namely, invariance under the transformation of $p$-thinning, followed by conditioning on being positive.

For each $\beta \in $, we show that there is a unique distribution which is a fixed point of this transformation; the distribution is power-law-$\beta$, and different from the usual Yule-Simon power law-$\beta$ that arises in preferential attachment models.

In addition to characterizing these fixed points, we prove convergence results for iterates of the transformation.

In this note we study some properties of infinite percolation clusters on non-amenable graphs. In particular, we study the percolative properties of the complement of infinite percolation clusters. An approach based on mass-transport is adapted to show that for a large class of non-amenable graphs, the graph obtained by removing each site contained in an infinite percolation cluster has critical percolation threshold which can be arbitrarily close to the critical threshold for the original graph, almost surely, as $p\searrow p_c$. Closely related is the self-destructive percolation process, introduced by J. van den Berg and R. Brouwer, for which we prove that an infinite cluster emerges for any small reinforcement.

We consider the probability that a weighted sum of n i.i.d. random variables $X_j, j = 1,\ldots,n$, with stretched exponential tails is larger than its expectation and determine the rate of its decay, under suitable conditions on the weights. We show that the decay is subexponential, and identify the rate function in terms of the tails of $X_j$ and the weights. Our result generalizes the large deviation principle given by Kiesel and Stadtmüller as well as the tail asymptotics for sums of i.i.d. random variables provided by Nagaev. As an application of our result, motivated by random projections of high-dimensional vectors, we consider the case of random, self-normalized weights that are independent of the sequence $X_j$, identify the decay rate for both the quenched and annealed large deviations in this case, and show that they coincide. As another example we consider weights derived from kernel functions that arise in nonparametric regression.

Bootstrap percolation is a cellular automaton modelling the spread of an `infection' on a graph. In this note, we prove a family lower bounds on the critical probability for r-neighbour bootstrap percolation on Galton-Watson trees in terms of moments of the offspring distributions. With this result we confirm a conjecture of Bollobás, Gunderson, Holmgren, Janson and Przykucki. We also show that these bounds are best possible up to positive constants not depending on the offspring distribution.

In this paper we prove the uniqueness of solutions to degenerate Fokker-Planck equations with bounded coefficients, under the additional assumptions that the diffusion coefficient has $W^{1,2}_{loc}$ regularity, while the gradient of the drift coefficient is merely given by a singular integral.

We prove almost sure convergence of the maximum degree in an evolving tree model combining local choice and preferential attachment. At each step in the growth of the graph, a new vertex is introduced. A fixed, finite number of possible neighbors are sampled from the existing vertices with probability proportional to degree. Of these possibilities, the new vertex attaches to the vertex from the sample that has the highest degree. The maximal degree in this model has linear or near-linear behavior. This behavior contrasts sharply with the behavior in the same choice model with uniform attachment as well as the preferential attachment model without choice. The proof is based on showing the tree has a persistent hub by comparison with the standard preferential attachment model, as well as martingale and stochastic approximation arguments.

We consider a d-dimensional SDE with an identity diffusion matrix and a drift vector being a vector function of bounded variation. We give a representation for the derivative of the solution with respect to the initial data.

We initiate the study of mixing times of Markov chain under monotone censoring. Suppose we have some Markov Chain $M$ on a state space $\Omega$ with stationary distribution $\pi$ and a monotone set $A \subset \Omega$. We consider the chain $M'$ which is the same as the chain $M$ started at some $x \in A$ except that moves of $M$ of the form $x \to y$ where $x \in A$ and $y \notin A$ are {\em censored} and replaced by the move $x \to x$. If $M$ is ergodic and $A$ is connected, the new chain converges to $\pi$ conditional on $A$. In this paper we are interested in the mixing time of the chain $M'$ in terms of properties of $M$ and $A$. Our results are based on new connections with the field of property testing. A number of open problems are presented.

A spectral decomposition for the generator and the transition probabilities of the block counting process of the Bolthausen-Sznitman coalescent is derived. This decomposition is closely related to the Stirling numbers of the first and second kind. The proof is based on generating functions and exploits a certain factorization property of the Bolthausen-Sznitman coalescent. As an application we derive a formula for the hitting probability $h(i,j)$ that the block counting process of the Bolthausen-Sznitman coalescent ever visits state $j$ when started from state $i\ge j$. Moreover, explicit formulas are derived for the moments and the distribution function of the absorption time $\tau_n$ of the Bolthausen-Sznitman coalescent started in a partition with $n$ blocks. We provide an elementary proof for the well known convergence of $\tau_n-\log\log n$ in distribution to the standard Gumbel distribution. It is shown that the speed of this convergence is of order $1/\log n$.

I consider a stochastic optimization problem for a one-dimensional continuous martingale whose diffusion rate is constrained to be between two positive values $r_{1}<r_{2}$. The problem is to find an optimal adapted strategy for the choice of diffusion rate in order to maximize the chance of hitting an infinitesimal region around the origin at a set time in the future. More precisely, the parameter associated with "the chance of hitting the origin" is the exponent for a singularity induced at the origin of the final time probability density. I show that the optimal exponent solves a transcendental equation depending on the ratio $\frac{r_{2}}{r_{1}}$.

In this paper, we establish the ergodicity of the Airy line ensemble with respect to horizontal shifts. This shows that it is the only candidate for Conjecture 3.2 in Corwin & Hammond, Invent. Math. 2014, regarding the classification of ergodic line ensembles satisfying a certain Brownian Gibbs property after a parabolic shift.

We give a short and elementary proof of a symmetry identity for the $q$-moments of the $q$-Hahn distribution arising in the study of the $q$-Hahn Boson process and the $q$-Hahn TASEP. This identity discovered by Corwin in "The q-Hahn Boson Process and q-Hahn TASEP", Int. Math. Res. Not., 2014, was a key technical step to prove an intertwining relation between the Markov transition matrices of these two classes of discrete-time Markov chains. This was used in turn to derive exact formulas for a large class of observables of both these processes.

We give a new proof of a result of Rick Kenyon that the probability that an edge in the middle of an $n \times n$ square is used in a loop-erased walk connecting opposite sides is of order $n^{-3/4}$. We, in fact, improve the result by showing that this estimate is correct up to multiplicative constants.

Consider bond percolation on the square lattice \(\mathbb{Z}^2\) where each edge is independently open with probability \(p.\) For some positive constants \(p_0 \in

The totally disconnectedness of support for super Brownian motion in high dimensions is well known. In this paper, we prove that similar results also hold for $\Lambda$-Fleming-Viot process with Brownian spatial motion provided that the associated $\Lambda$-coalescent does not come down from infinity fast enough. Our proof is another application of the lookdown particle representation for $\Lambda$-Fleming-Viot process. We also discuss the disjointness of independent $\Lambda$-Fleming-Viot supports and ranges in high dimensions. The disconnectedness of the $\Lambda$-Fleming-Viot support remains open in certain low dimensions.

We use a growth procedure for binary trees due to Luczak and Winkler, a bijection between binary trees and irreducible quadrangulations of the hexagon due to Fusy, Poulalhon and Schaeffer, and the classical angular mapping between quadrangulations and maps, to define a growth procedure for maps. The growth procedure is local, in that every map is obtained from its predecessor by an operation that only modifies vertices lying on a common face with some fixed vertex. As n tends to infinity, the probability that the n'th map in the sequence is 3-connected tends to 2^8/3^6. The sequence of maps has an almost sure limit G, and we show that G is the distributional local limit of large, uniformly random 3-connected graphs.

A tight upper bound is given on the distribution of the maximum of a supermartingale. Specifically, it is shown that if $Y$ is a semimartingale with initial value zero and quadratic variation process $[Y,Y]$ such that $Y + [Y,Y]$ is a supermartingale, then the probability the maximum of $Y$ is greater than or equal to a positive constant $a$ is less than or equal to$1/(1+a).$ The proof makes use of the semimartingale calculus and is inspired by dynamic programming.

We investigate analytical properties of free stable distributions and discover many connections with their classical counterparts. Our main result is an explicit formula for the Mellin transform, which leads to explicit series representations for the characteristic function and for the density of a free stable distribution. All of these formulas bear close resemblance to the corresponding expressions for classical stable distributions. As further applications of our results, we give an alternative proof of the duality law due to Biane and a new factorization of a classical stable random variable into an independent (in the classical sense) product of a free stable random variable and a power of a Gamma(2) random variable.

We give an explicit bound for the $L_1$-distance between two additive processes of local characteristics $(f_j(\cdot),\sigma^2(\cdot),\nu_j)$, $j = 1,2$. The cases $\sigma =0$ and $\sigma(\cdot) > 0$ are both treated. We allow $\nu_1$ and $\nu_2$ to be time-homogeneous Lévy measures, possibly with infinite variation. Some examples of possible applications are discussed.<br /><br />

Typically, a stochastic model relates stochastic “inputs” and, perhaps, controls tostochastic “outputs”. A general version of the Yamada-Watanabe and Engelbert the-orems relating existence and uniqueness of weak and strong solutions of stochasticequations is given in this context. A notion of compatibility between inputs and out-puts is critical in relating the general result to its classical forebears. The usualformulation of stochastic differential equations driven by semimartingales does notrequire compatibility, so a notion of partial compatibility is introduced which doeshold. Since compatibility implies partial compatibility, classical strong uniquenessresults imply strong uniqueness for compatible solutions. Weak existence argumentstypically give existence of compatible solutions (not just partially compatible solu-tions), and as in the original Yamada-Watanabe theorem, existence of strong solutionsfollows.

We provide a streamlined proof and improved estimates for the weak multivariate Gnedenko law of large numbers on concentration of random polytopes within the space of convex bodies (in a fixed or a high dimensional setting), as well as a corresponding strong law of large numbers.

For a strictly stationary sequence of regularly varying random variables we study functional weak convergence of partial sum processes in the space $D[0,1]$ with the $J_{1}$ topology. Under the strong mixing condition, we identify necessary and sufficient conditions for such convergence in terms of the corresponding extremal index. We also give conditions under which the regular variation property is a necessary condition for this functional convergence in the case of weak dependence.

We reveal a connection of the Brascamp-Lieb inequality with Skorokhod embedding. Error bounds for the inequality in terms of the variance are also provided.

We show that in a separable infinite dimensional Hilbert space, uniform integrability of the square of the norm of normalized partial sums of a strictly stationary sequence, together with a strong mixing condition, does not guarantee the central limit theorem.

The aim of this paper is to investigate the Gibbs sampling that's used for computing the mean of observables with respect to some function $f$ depending on a very small number of variables. For this type of observable, by using the $d_{l_{2}}$-metric one obtains the sharp concentration estimate for the empirical mean, which in particular yields the correct speed in the concentration for $f$ depending on a single observable.

In this paper we consider the winding number, $\theta(s)$, of planar Brownian motion and study asymptotic behavior of the process of the maximum time, the time when $\theta(s)$ attains the maximum in the interval $0\le s \le t$. We find the limit law of its logarithm with a suitable normalization factor and the upper growth rate of the maximum time process itself. We also show that the process of the last zero time of $\theta(s)$ in $[0,t]$ has the same law as the maximum time process.

Petrov constructed a diffusion process in the Kingman simplex whose unique stationary distribution is the two-parameter Poisson-Dirichlet distribution of Pitman and Yor. We show that the subset of the simplex comprising vectors whose coordinates sum to 1 is the natural state space for the process. In fact, the complementary set acts like an entrance boundary.

Let $f:\mathbb{R}^k\to\mathbb{R}$ be a measurable function, and let ${(U_i)}_{i\in\mathbb{N}}$ be a sequence of i.i.d. random variables. Consider the random process $Z_i=f(U_{i},...,U_{i+k-1})$. We show that for all $\ell$, there is a positive probability, uniform in $f$, for $Z_1,...,Z_\ell$ to be monotone. We give upper and lower bounds for this probability, and draw corollaries for $k$-block factor processes with a finite range. The proof is based on an application of combinatorial results from Ramsey theory to the realm of continuous probability.

We consider a special case of the generalized Pólya's urn model. Given a finite connected graph $G$, place a bin at each vertex. Two bins are called a pair if they share an edge of $G$. At discrete times, a ball is added to each pair of bins. In a pair of bins, one of the bins gets the ball with probability proportional to its current number of balls. A question of essential interest for the model is to understand the limiting behavior of the proportion of balls in the bins for different graphs $G$. In this paper, we present two results regarding this question. If $G$ is not balanced-bipartite, we prove that the proportion of balls converges to some deterministic point $v=v(G)$ almost surely. If $G$ is regular bipartite, we prove that the proportion of balls converges to a point in some explicit interval almost surely. The question of convergence remains open in the case when $G$ is non-regular balanced-bipartite.

We show that the splitting-characterization of the Poisson point process is an immediate consequence of the Mecke-formula.

Large deviations for the local time of a process $X_i$ are investigated, where $X_i=x_i$ for $t∈[S_{i-1},S_i[$ and $(x_j)$ are i.i.d. random variables on a Polish space, S_j is the $j$-th arrival time of a renewal process depending on $(x_j)$. No moment conditions are assumed on the arrival times of the renewal process.

Let $\{X_n\}_{n\geq1}$ be a sequence of i.i.d. standard Gaussian random variables, let $S_n=\sum_{i=1}^nX_i$ be the Gaussian random walk, and let $T_n=\sum_{i=1}^nS_i$ be the integrated (or iterated) Gaussian random walk. In this paper we derive the following upper and lower bounds for the conditional persistence:$$\mathbb{P}\left\{\max_{1\leq k \leq n}T_{k} \leq 0\,\,\Big|\,\,T_n=0,S_n=0\right\}\lesssim n^{-1/2},\ \mathbb{P}\left\{\max_{1\leq k \leq 2n}T_{k} \leq 0\,\,\Big|\,\,T_{2n}=0,S_{2n}=0\right\}\gtrsim\frac{n^{-1/2}}{\log n},$$for $n\rightarrow\infty,$ which partially proves a conjecture by Caravenna and Deuschel (2008).

In the paper "Transience and recurrence of rotor-router walks on directed covers of graphs", published in <a href="http://dx.doi.org/10.1214/ECP.v17-2096">ECP volume 17 (2012), no. 41</a> there is an error in the proof of Corollary 3.8. This corollary is essential for the transient part in the proof of Theorem 3.5(b). We fix this error by constructing a new rotor-router process, which fulfills our needs, and for which the statement of Corollary 3.8. holds.

We give a multivariate generalization of Borell's noise stability theorem for Gaussian vectors. As a consequence we recover two inequalities, also due to Borell, for exit times of the Ornstein-Uhlenbeck process.

We apply the stochastic Perron method of Bayraktar and Sîrbu to a general infinite horizon optimal control problem, where the state $X$ is a controlled diffusion process, and the state constraint is described by a closed set. We prove that the value function $v$ is bounded from below (resp., from above) by a viscosity supersolution (resp., subsolution) of the related state constrained problem for the Hamilton-Jacobi-Bellman equation. In the case of a smooth domain, under some additional assumptions, these estimates allow to identify $v$ with a unique continuous constrained viscosity solution of this equation.

We continue the line of research of random walks with a barrier initiated by Iksanov and Möhle (2008). Assuming that the tail of the step of the underlying random walk has a power-like behavior at infinity with the exponent $-\alpha$, $\alpha\in.

A natural extension of a right-continuous integer-valued random walk is one which can jump to the right by one or two units. First passage times above a given fixed level then admit - on each of the two events, which correspond to overshoot zero and one, separately - a tractable probability generating function. Some applications are considered.

We consider recurrence versus transience for models of random walks on growing in time, connected subsets $\mathbb{G}_t$ of some fixed locally finite, connected graph, in which monotone interaction enforces such growth as a result of visits by the walk (or probes it sent), to the neighborhood of the boundary of $\mathbb{G}_t$.

We consider a sequence of Markov chains $(\mathcal X^n)_{n=1,2,...}$ with $\mathcal X^n = (X^n_\sigma)_{\sigma\in\mathcal T}$, indexed by the full binary tree $\mathcal T = \mathcal T_0 \cup \mathcal T_1 \cup ...$, where $\mathcal T_k$ is the $k$th generation of $\mathcal T$. In addition, let $(\Sigma_k)_{k=0,1,2,...}$ be a random walk on $\mathcal T$ with $\Sigma_k \in \mathcal T_k$ and $\widetilde{\mathcal R}^n = (\widetilde R_t^n)_{t\geq 0}$ with $\widetilde R_t^n := X_{\Sigma_{[tn]}}$, arising by observing the Markov chain $\mathcal X^n$ along the random walk. We present a law of large numbers concerning the empirical measure process $\widetilde{\mathcal Z}^n = (\widetilde Z_t^n)_{t\geq 0}$ where $\widetilde{Z}_t^n = \sum_{\sigma\in\mathcal T_{[tn]}} \delta_{X_\sigma^n}$ as $n\to\infty$. Precisely, we show that if $\widetilde{\mathcal R}^n \Rightarrow{n\to\infty} \mathcal R$ for some Feller process $\mathcal R = (R_t)_{t\geq 0}$ with deterministic initial condition, then $\widetilde{\mathcal Z}^n \Rightarrow{n\to\infty} \mathcal Z$ with $Z_t = \delta_{\mathcal L(R_t)}$.

Let $X_1,X_2,\ldots$ be independent identically distributed nonnegative random variables. Wald's identity states that the random sum $S_T:=X_1+\cdots+X_T$ has expectation $\mathbb{E} T \cdot \mathbb{E} X_1$ provided $T$ is a stopping time. We prove here that for any $1<\alpha\leq 2$, if $T$ is an arbitrary nonnegative random variable, then $S_T$ has finite expectation provided that $X_1$ has finite $\alpha$-moment and $T$ has finite $1/(\alpha-1)$-moment. We also prove a variant in which $T$ is assumed to have a finite exponential moment. These moment conditions are sharp in the sense that for any i.i.d. sequence $X_i$ violating them, there is a $T$ satisfying the given condition for which $S_T$ (and, in fact, $X_T$) has infinite expectation.An interpretation of this is given in terms of a prophet being more rewarded than a gambler when a certain impatience restriction is imposed.

In this paper we present a recurrence criterion for the frog model on $\mathbb{Z}^d$ with an i.i.d. initial configuration of sleeping frogs and such that the underlying random walk has a drift to the right.

We prove the existence of uncountably many nonnegative harmonic functions for random walks in the euclidean space with non-zero drift, killed when leaving general convex cones with vertex in 0. We also make the natural conjecture about the Martin boundary for lattice random walks in general convex cones in two dimensions. Proving that the set of harmonic functions found is the full Martin boundary for these processes is an open problem.

We consider a Markovian stochastic control problem with model uncertainty. The controller (intelligent player) observes only the state, and, therefore, uses feedback (closed-loop) strategies. The adverse player (nature) who does not have a direct interest in the payoff, chooses open-loop controls that parametrize Knightian uncertainty. This creates a two-step optimization problem (like half of a game) over feedback strategies and open-loop controls. The main result is to show that, under some assumptions, this provides the same value as the (half of) the zero-sum symmetric game where the adverse player also plays feedback strategies and actively tries to minimize the payoff. The value function is independent of the filtration accessible to the adverse player. Aside from the modeling issue, the present note is a technical companion to a previous work.

We show that in the Schrödinger point process, Sch$_\tau$, $\tau>0,$ the probability of having no eigenvalue in a fixed interval of size $\lambda$ is given by \[ \exp\left(-\frac{\lambda^2}{4\tau}+\left(\frac{2}{\tau}-\frac{1}{4}\right)\lambda +o(\lambda)\right), \]as $\lambda\to\infty.$ It is a slightly more precise version than the one given in a previous work.

We provide tight lower bounds on the smallest eigenvalue of a sample covariance matrix of a centred isotropic random vector under weak or no assumptions on its components.

In this note we look into detail into the box-counting dimension of subordinators. Given that X is a non-decreasing Levy process which is not Compound Poisson process we show that in the limit, a.s., the minimum number of boxes of size $a$ that cover the range of $(X_s)_{s\leq t}$ is a.s. of order $t/U(a)$, where U is the potential function of X. This is a more rened result than the lower and upper index of the box-counting dimension computed by Jean Bertoin in his 1999 book, which deals with the asymptotic of the number of boxes at logarithmic scale.

The theory of $S_2(\delta)$ family of probability distributions is used to give a derivation of the functional equation of the Riemann xi function. The $\delta$ deformation of the xi function is formulated in terms of the $S_2(\delta)$ distribution and shown to satisfy Riemann's functional equation. Criteria for simplicity of roots of the xi function and for its simple roots to satisfy the Riemann hypothesis are formulated in terms of a differentiability property of the $S_2(\delta)$ family. For application, the values of the Riemann zeta function at the integers and of the Riemann xi function in the complex plane are represented as integrals involving the Laplace transform of $S_2.$

A representation is given for a large class of n-dimensional multivariate gamma random variables as defined by Verre-Jones. In particular, the probability density functions of all 2-dimensional gamma random variables are given explicitly and it is shown how to obtain the probability density functions of all 3-dimensional gamma random variables.

In the case of neutral populations of fixed sizes in equilibrium whose genealogies are described by the Kingman N-coalescent back from time t consider the associated processes of total tree length as t increases. We show that the (càdlàg) process to which the sequence of compensated tree length processes converges as N tends to infinity is a process of infinite quadratic variation; therefore this process cannot be a semimartingale. This answers a question posed in Pfaffelhuber et al. (2011).