We prove that an optional process of non-exploding realized power variation along stopping times possesses almost surely làglàd paths. This result is useful for the analysis of some imperfect market models in mathematical finance. In the finance applications variation naturally appears along stopping times and not pathwise. On the other hand, if the power variation were only taken along deterministic points in time, the assertion would obviously be wrong.

Let $X$, $Y$ be continuous-time martingales taking values in a separable Hilbert space $\mathcal{H}$.

(i) Assume that $X$, $Y$ satisfy the condition $[X,X]_t\geq [Y,Y]_t$ for all $t\geq 0$. We prove the sharp inequalities $$ \sup_t||Y_t||_p\leq (p-1)^{-1}\sup_t||X_t||_p,\qquad 1 < p\leq 2,$$ $$ \mathbb{P}(\sup_t|Y_t|\geq 1)\leq \frac{2}{\Gamma(p+1)}\sup_t||X_t||_p^p,\qquad 1\leq p\leq 2,$$ and for any $K>0$ we determine the optimal constant $L=L(K)$ depending only on $K$ such that $$ \sup_t ||Y_t||_1\leq K\sup_t\mathbb{E}|X_t|\log|X_t|+L(K).$$

(ii) Assume that $X$, $Y$ satisfy the condition $[X,X]_\infty-[X,X]_{t-}\geq [Y,Y]_\infty-[Y,Y]_{t-}$ for all $t\geq 0$. We establish the sharp bounds $$ \sup_t||Y_t||_p\leq (p-1)\sup_t||X_t||_p,\qquad 2\leq p < \infty$$ and $$ \mathbb{P}(\sup_t|Y_t|\geq 1)\leq \frac{p^{p-1}}{2}\sup_t||X_t||_p^p,\qquad 2\leq p < \infty.$$

This generalizes the previous results of Burkholder, Suh and the author, who showed the above estimates under the more restrictive assumption of differential subordination. The proof is based on Burkholder's technique and integration method.

For first passage percolation in $\mathbb{Z}^d$ with large $d$, we construct a path connecting the origin to $\{x_1 =1\}$, whose passage time has optimal order $\log d/d$. Besides, an improved lower bound for the "diagonal" speed of the cluster combined with a result by Dhar (1988) shows that the limiting shape in FPP with exponential passage times (and thus that of Eden model) is not the euclidean ball in dimension larger than 35.

In a one-parameter model for evolution of random trees, which also includes the Barabasi-Albert random tree [1], almost sure behavior and the limiting distribution of the degree of a vertex in a fixed position are examined. A functional central limit theorem is also given. Results about Polya urn models are applied in the proofs.

We consider the complex case of the S-inequality. It concerns the behaviour of Gaussian measures of dilations of convex and rotationally symmetric sets in $\mathbb{C}^n$. We pose and discuss a conjecture that among all such sets measures of cylinders decrease the fastest under dilations. Our main result in this paper is that this conjecture holds under the additional assumption that the Gaussian measure of the sets considered is not greater than some constant $c > 0.64$.

We introduce and study a new interacting particles model with a wall and two kinds of interactions - blocking and pushing - which maintain particles in a certain order. We show that it involves a random matrix model.

Let $Y$ be a nonnegative random variable with mean $\mu$ and finite positive variance $\sigma^2$, and let $Y^s$, defined on the same space as $Y$, have the $Y$ size biased distribution, that is, the distribution characterized by $$ E[Yf(Y)]=\mu E f(Y^s) \quad \mbox{for all functions $f$ for which these expectations exist.} $$ Under a variety of conditions on the coupling of $Y$ and $Y^s$, including combinations of boundedness and monotonicity, concentration of measure inequalities such as $$ P\left(\frac{Y-\mu}{\sigma}\ge t\right)\le \exp\left(-\frac{t^2}{2(A+Bt)}\right) \quad \mbox{for all $t \ge 0$} $$ are shown to hold for some explicit $A$ and $B$ in \cite{cnm}. Such concentration of measure results are applied to a number of new examples: the number of relatively ordered subsequences of a random permutation, sliding window statistics including the number of $m$-runs in a sequence of coin tosses, the number of local maxima of a random function on a lattice, the number of urns containing exactly one ball in an urn allocation model, and the volume covered by the union of $n$ balls placed uniformly over a volume $n$ subset of $\mathbb{R}^d$.

We study the density of the supremum of a strictly stable Levy process. We prove that for almost all values of the index $\alpha$ - except for a dense set of Lebesgue measure zero - the asymptotic series which were obtained in Kuznetsov (2010) "On extrema of stable processes" are in fact absolutely convergent series representations for the density of the supremum.

In this paper, we generalize the asymptotic result of Esseen (1958) concerning the Wasserstein distance of order one in the mean central limit theorem to the Wasserstein distances of order $r$ for $r \in ]1,2]$.

In this note, we revisit the work of T. Tao and V. Vu on large non-hermitian random matrices with independent and identically distributed (i.i.d.) entries with mean zero and unit variance. We prove under weaker assumptions that the limit spectral distribution of sum and product of non-hermitian random matrices is universal. As a byproduct, we show that the generalized eigenvalues distribution of two independent matrices converges almost surely to the uniform measure on the Riemann sphere.

We consider the maximum of the discrete two dimensional Gaussian free field in a box, and prove the existence of a (dense) deterministic subsequence along which the maximum, centered at its mean, is tight. The method of proof relies on an argument developed by Dekking and Host for branching random walks with bounded increments and on comparison results specific to Gaussian fields.

In this note, we provide an innovative and simple approach for proving the existence of a unique solution for multidimensional reflected BSDEs associated to switching problems. Getting rid of a monotonicity assumption on the driver function, this approach simplifies and extends the recent results of Hu and Tang (2008) or Hamadene and Zhang (2010).

We prove that the growth constants for nearest-neighbour lattice trees and lattice (bond) animals on the integer lattice $\mathbb{Z}^d$ are asymptotic to $2de$ as the dimension goes to infinity, and that their critical one-point functions converge to $e$. Similar results are obtained in dimensions $d > 8$ in the limit of increasingly spread-out models; in this case the result for the growth constant is a special case of previous results of M. Penrose. The proof is elementary, once we apply previous results of T. Hara and G. Slade obtained using the lace expansion.

According to a representation due to M. Kanter, the density of some power of a positive stable distribution is a completely monotone function. In this paper, we first derive its representative Bernstein measure which also describes the law of some function of a uniform random variable, referred to below as the Kanter random variable. Then, the distribution function of the latter variable is written down and gives a more explicit description of the non commutative analogue of positive stable distributions in the setting of Voiculescu's free probability theory. Analytic evidences of the occurrence of the Kanter random variable in both the classical and the free settings conclude the exposition.

This paper establishes Fokker-Planck-Kolmogorov type equations for time-changed Gaussian processes. Examples include those equations for a time-changed fractional Brownian motion with time-dependent Hurst parameter and for a time-changed Ornstein-Uhlenbeck process. The time-change process considered is the inverse of either a stable subordinator or a mixture of independent stable subordinators.

Using the framework of random walks in random scenery, Cohen and Samorodnitsky (2006) introduced a family of symmetric $\alpha$-stable motions called local time fractional stable motions. When $\alpha=2$, these processes are precisely fractional Brownian motions with $1/2 < H < 1$. Motivated by random walks in alternating scenery, we find a complementary family of symmetric $\alpha$-stable motions which we call indicator fractional stable motions. These processes are complementary to local time fractional stable motions in that when $\alpha=2$, one gets fractional Brownian motions with $0 < H < 1/2$.

A marked metric measure space (mmm-space) is a triple $(X,r,μ)$, where $(X,r)$ is a complete and separable metric space and $μ$ is a probability measure on $X \times I$ for some Polish space $I$ of possible marks. We study the space of all (equivalence classes of) marked metric measure spaces for some fixed $I$. It arises as a state space in the construction of Markov processes which take values in random graphs, e.g. tree-valued dynamics describing randomly evolving genealogical structures in population models. We derive here the topological properties of the space of mmm-spaces needed to study convergence in distribution of random mmm-spaces. Extending the notion of the Gromov-weak topology introduced in (Greven, Pfaffelhuber and Winter, 2009), we define the marked Gromov-weak topology, which turns the set of mmm-spaces into a Polish space. We give a characterization of tightness for families of distributions of random mmm-spaces and identify a convergence determining algebra of functions, called polynomials.

We correct an error in Electronic Communications in Probability, Vol 15 (2010), paper 10

We present plane algebraic curves that have segments of points for which non uniform self-similar measures get singular. We calculate appropriate points on the curves using Mathematica. These points are in the parameter domain where we generically have absolute continuity of the measures,

A class of integrals with respect to homogeneous Lévy bases on $\mathbb{R}^k$ is considered. In the one-dimensional case $k=1$ this class corresponds to the selfdecomposable distributions. Necessary and sufficient conditions for existence as well as some representations of the integrals are given. Generalizing the one-dimensional case it is shown that the class of integrals corresponds to Urbanik's class $ L_{k-1}(R)$. Finally, multiparameter Ornstein-Uhlenbeck processes are defined and studied.

This article studies the behavior of stochastic reaction-diffusion equations driven by additive regularly varying pure jump L'evy noise in the limit of small noise intensity. It is shown that the law of the suitably renormalized first exit times from the domain of attraction of a stable state converges to an exponential law of parameter 1 in a strong sense of Laplace transforms, including exponential moments. As a consequence, the expected exit times increase polynomially in the inverse intensity, in contrast to Gaussian perturbations, where this growth is known to be of exponential rate.

In this paper we provide some sharp asymptotic results for a stochastic model of species survival recently proposed by Guiol, Machado, and Schinazi.

We prove the uniqueness of the martingale problem associated to some degenerate operators. The key point is to exploit the strong parallel between the new technique introduced by Bass and Perkins [BP09] to prove uniqueness of the martingale problem in the framework of non- degenerate elliptic operators and the Mc Kean and Singer [MS67] parametrix approach to the density expansion that has previously been extended to the degenerate setting that we consider (see Delarue and Menozzi [DM10]).

Consider a Markov chain $(\xi_v)_{v \in V} \in [k]^V$ on the infinite $b$-ary tree $T = (V,E)$ with irreducible edge transition matrix $M$, where $b \geq 2$, $k \geq 2$ and $[k] = \{1,\ldots,k\}$. We denote by $L_n$ the level-$n$ vertices of $T$. Assume $M$ has a real second-largest (in absolute value) eigenvalue $\lambda$ with corresponding real eigenvector $\nu \neq 0$. Letting $\sigma_v = \nu_{\xi_v}$, we consider the following root-state estimator, which was introduced by Mossel and Peres (2003) in the context of the ``recontruction problem'' on trees: \begin{equation*} S_n = (b\lambda)^{-n} \sum_{x\in L_n} \sigma_x. \end{equation*} As noted by Mossel and Peres, when $b\lambda^2 > 1$ (the so-called Kesten-Stigum reconstruction phase) the quantity $S_n$ has uniformly bounded variance. Here, we give bounds on the moment-generating functions of $S_n$ and $S_n^2$ when $b\lambda^2 > 1$. Our results have implications for the inference of evolutionary trees.

Freedman's inequality is a martingale counterpart to Bernstein's inequality. This result shows that the large-deviation behavior of a martingale is controlled by the predictable quadratic variation and a uniform upper bound for the martingale difference sequence. Oliveira has recently established a natural extension of Freedman's inequality that provides tail bounds for the maximum singular value of a matrix-valued martingale. This note describes a different proof of the matrix Freedman inequality that depends on a deep theorem of Lieb from matrix analysis. This argument delivers sharp constants in the matrix Freedman inequality, and it also yields tail bounds for other types of matrix martingales. The new techniques are adapted from recent work by the present author.

Branching annihilating random walk (BARW) is a generic term for a class of interacting particle systems on $\mathbb{Z}^d$ in which, as time evolves, particles execute random walks, produce offspring (on neighbouring sites) and (instantaneously) disappear when they meet other particles. Much of the interest in such models stems from the fact that they typically lack a monotonicity property called attractiveness, which in general makes them exceptionally hard to analyse and in particular highly sensitive in their qualitative long-time behaviour to even slight alterations of the branching and annihilation mechanisms. In this short note, we introduce so-called caring double-branching annihilating random walk (cDBARW) on $\mathbb{Z}$, and investigate its long-time behaviour. It turns out that it either allows survival with positive probability if the branching rate is greater than $1/2$, or a.s. extinction if the branching rate is smaller than $1/3$ and (additionally) branchings are only admitted for particles which have at least one neighbouring particle (so-called 'cooperative branching'). Further, we show a.s. extinction for all branching rates for a variant of this model, where branching is only allowed if offspring can be placed at odd distance between each other. It is the latter (extinction-type) results which seem remarkable, since they appear to hint at a general extinction result for a non-trivial parameter range in the so-called 'parity-preserving universality class', suggesting the existence of a 'true' phase transition. The rigorous proof of such a non-trivial phase transition remains a particularly challenging open problem.

In this paper, we provide the sharp asymptotics for the quantization radius (maximal radius) for a sequence of optimal quantizers for random variables $X$ in $(\mathbb{R}^d,\|\,\cdot\,\|)$ with radial exponential tails. This result sharpens and generalizes the results developed for the quantization radius in [4] for $d > 1$, where the weak asymptotics is established for similar distributions in the Euclidean case. Furthermore, we introduce quantization balls, which provide a more general way to describe the asymptotic geometric structure of optimal codebooks, and extend the terminology of the quantization radius.

We consider moments of the return times (or first hitting times) in an irreducible discrete time discrete space Markov chain. It is classical that the finiteness of the first moment of a return time of one state implies the finiteness of the first moment of the first return time of any other state. We extend this statement to moments with respect to a function $f$, where $f$ satisfies a certain, best possible condition. This generalizes results of K.L. Chung (1954) who considered the functions $f(n)=n^p$ and wondered "[...] what property of the power $n^p$ lies behind this theorem [...]" (see Chung (1967), p. 70). We exhibit that exactly the functions that do not increase exponentially - neither globally nor locally - fulfill the above statement.

We prove the existence of a solution for the Kolmogorov equation associated with a reflection problem for 2-D stochastic Navier-Stokes equations with periodic spatial conditions and the corresponding stream flow in a closed ball of a Sobolev space of the torus $T^2$.

The purpose of this note is to establish a Central Limit Theorem for the number of eigenvalues of a Wigner matrix in an interval. The proof relies on the correct asymptotics of the variance of the eigenvalue counting function of GUE matrices due to Gustavsson, and its extension to large families of Wigner matrices by means of the Tao and Vu Four Moment Theorem and recent localization results by Erd?s, Yau and Yin.

We prove a localization-result for branching random walks in random environment, namely that if the process does not die out, the most populated site will infinitely often contain more than a fixed percentage of the population. This had been proven already before by Hu and Yoshida, but it is possible to drop their assumption that particles may not die.

We consider the white-noise driven stochastic heat equation on $[0,1]$ with Lipschitz-continuous drift and diffusion coefficients. We derive an inequality for the $L^1([0,1])$-norm of the difference between two solutions. Using some martingale arguments, we show that this inequality provides some estimates which allow us to study the stability of the solution with respect the initial condition, the uniqueness of the possible invariant distribution and the asymptotic confluence of solutions.

In this work we address the problem of determining the asymptotic spectral measure of the product of independent, Gaussian random matrices with correlated entries, as the dimension and the number of multiplicative terms goes to infinity. More specifically, let $\{X_p(N)\}_{p=1}^\infty$ be a sequence of $N\times N$ independent random matrices with independent and identically distributed Gaussian entries of zero mean and variance $\frac{1}{\sqrt{N}}$. Let $\{\Sigma(N)\}_{N=1}^\infty$ be a sequence of $N\times N$ deterministic and Hermitian matrices such that the sequence converges in moments to a compactly supported probability measure $\sigma$. Define the random matrix $Y_p(N)$ as $Y_p(N)=X_p(N)\Sigma(N)$. This is a random matrix with correlated Gaussian entries and covariance matrix $E(Y_p(N)^*Y_p(N))=\Sigma(N)^2$ for every $p\geq 1$. The positive definite $N\times N$ matrix $$ B_n^{1/(2n)} (N) := \left( Y_1^* (N) Y_2^* (N) \dots Y_n^*(N) Y_n(N) \dots Y_2(N) Y_1(N) \right)^{1/(2n)} \to \nu_n $$ converges in distribution to a compactly supported measure in $[0,\infty)$ as the dimension of the matrices $N\to \infty$. We show that the sequence of measures $\nu_n$ converges in distribution to a compactly supported measure $\nu_n \to \nu$ as $n\to\infty$. The measures $\nu_n$ and $\nu$ only depend on the measure $\sigma$. Moreover, we deduce an exact closed-form expression for the measure $\nu$ as a function of the measure $\sigma$.

In this paper we discuss the following problem: given a random variable $Z=X+Y$ with Gamma law such that $X$ and $Y$ are independent, we want to understand if then $X$ and $Y$ each follow a Gamma law. This is related to Cramer's theorem which states that if $X$ and $Y$ are independent then $Z=X+Y$ follows a Gaussian law if and only if $X$ and $Y$ follow a Gaussian law. We prove that Cramer's theorem is true in the Gamma context for random variables living in a Wiener chaos of fixed order but the result is not true in general. We also give an asymptotic variant of our result.

We consider the interlacement Poisson point process on the space of doubly-infinite $\mathbb{Z}^d$-valued trajectories modulo time-shift, tending to infinity at positive and negative infinite times. The set of vertices and edges visited by at least one of these trajectories is the graph induced by the random interlacements at level $u$ of Sznitman(2010). We prove that for any $u > 0$, almost surely, the random interlacement graph is transient.

We consider the one-sided exit problem for fractional Brownian motion (FBM), which is equivalent to the question of the distribution of the lower tail of the maximum of FBM on the unit interval. We improve the bounds given by Molchan (1999) and shed some light on the relation to the quantity I studied there.

The notion of half independence arises in random matrices and quantum groups. This notion is available only for elements of a noncommutative probability space and assumes the existence of all moments. We relate half independence to a certain class of partitions and use it to define an appropriate cumulant generating function and a transform which is closely related to the characteristic function. This leads to a definition of half independent convolution of arbitrary probability measures which is compatible with the distribution of the sum of half independent elements of a noncommutative probability space. We also establish the central limit theorem for half independent convolution of measures with the limit being symmetrized Rayleigh. Cramer's theorem is also established in this set up.

This article provides an overview of recent work on descriptions and properties of the Convex minorants of random walks and Lévy processes, which summarize and extend the literature on these subjects. The results surveyed include point process descriptions of the convex minorant of random walks and Lévy processes on a fixed finite interval, up to an independent exponential time, and in the infinite horizon case. These descriptions follow from the invariance of these processes under an adequate path transformation. In the case of Brownian motion, we note how further special properties of this process, including time-inversion, imply a sequential description for the convex minorant of the Brownian meander.

We consider multitype branching processes arising in the study of random laminations of the disk. We classify these processes according to their subcritical or supercritical behavior and provide Kolmogorov-type estimates in the critical case corresponding to the random recursive lamination process of [1]. The proofs use the infinite dimensional Perron-Frobenius theory and quasi-stationary distributions.

In this paper, a new decay estimate for a class of stochastic evolution equations with weakly dissipative drifts is established, which directly implies the uniqueness of invariant measures for the corresponding transition semigroups. Moreover, the existence of invariant measures and the convergence rate of corresponding transition semigroup to the invariant measure are also investigated. As applications, the main results are applied to singular stochastic $p$-Laplace equations and stochastic fast diffusion equations, which solves an open problem raised by Barbu and Da Prato in [Stoc. Proc. Appl. 120(2010), 1247-1266].

We analyze the tail behavior of the maximum $N$ of $\{W(t)-t^2:t\ge0\}$, where $W$ is standard Brownian motion on $[0,\infty)$, and give an asymptotic expansion for ${\mathbb P}\{N\ge x\}$, as $x\to\infty$. This extends a first order result on the tail behavior, which can be deduced from Hüsler and Piterbarg (1999). We also point out the relation between certain results in Janson et al. (2010) and Groeneboom (2010).

In 2005, Nualart and Peccati showed that, surprisingly, the convergence in distribution of a normalized sequence of multiple Wiener-Itô integrals towards a standard Gaussian law is equivalent to convergence of just the fourth moment to 3. Recently, this result is extended to a sequence of multiple Wigner integrals, in the context of free Brownian motion. The goal of the present paper is to offer an elementary, unifying proof of these two results. The only advanced, needed tool is the product formula for multiple integrals. Apart from this formula, the rest of the proof only relies on soft combinatorial arguments.

We prove a law of large numbers for the right boundary in the model of internal DLA generated by cookie random walks in dimension one. The proof is based on stochastic recursions techniques.

Many kinds of independence have been defined in non-commutative probability theory. Natural independence is an important class of independence; this class consists of five independences (tensor, free, Boolean, monotone and anti-monotone ones). In the present paper, a unified treatment of joint cumulants is introduced for natural independence. The way we define joint cumulants enables us not only to find the monotone joint cumulants but also to give a new characterization of joint cumulants for other kinds of natural independence, i.e., tensor, free and Boolean independences. We also investigate relations between generating functions of moments and monotone cumulants. We find a natural extension of the Muraki formula, which describes the sum of monotone independent random variables, to the multivariate case.

It is shown that a stochastic process obtained by taking random sums of dilations and translations of a given function converges to Gaussian white noise if a dilation parameter grows to infinity and that it converges to Gaussian colored noise if a scaling parameter for the translations grows to infinity. In particular, the question of when one obtains fractional Brownian motion by integrating this colored noise is studied.

The directed last-passage percolation (LPP) model with independent exponential times is considered. We complete the study of asymptotic directions of infinite geodesics, started by Ferrari and Pimentel [5]. In particular, using a recent result of [3] and a local modification argument, we prove there is no (random) direction with more than two geodesics with probability 1.

We consider the geometry of random interlacements on the $d$-dimensional lattice. We use ideas from stochastic dimension theory developed in [1] to prove the following: Given that two vertices $x,y$ belong to the interlacement set, it is possible to find a path between $x$ and $y$ contained in the trace left by at most $\lceil d/2 \rceil$ trajectories from the underlying Poisson point process. Moreover, this result is sharp in the sense that there are pairs of points in the interlacement set which cannot be connected by a path using the traces of at most $\lceil d/2 \rceil-1$ trajectories.

Construction methods for Feller processes which require bounded coefficients are extended to the case of unbounded coefficients.

For four types of functions $\xi : ]0,\infty[\to ]0,\infty[$, we characterize the law of two independent and positive r.v.'s $X$ and $Y$ such that $U:=\xi(X+Y)$ and $V:=\xi(X)-\xi(X+Y)$ are independent. The case $\xi(x)=1/x$ has been treated by Letac and Wesolowski (2000). As for the three other cases, under the weak assumption that $X$ and $Y$ have density functions whose logarithm is locally integrable, we prove that the distribution of $(X,Y)$ is unique. This leads to Kummer, gamma and beta distributions. This improves the result obtained in [1] where more regularity was required from the densities.

A particular subclass of compound Poisson population models is analyzed. The models in the domain of attraction of the Kingman coalescent are characterized and it is shown that these models are never in the domain of attraction of any other continuous-time coalescent process. Results are obtained characterizing which of these models are in the domain of attraction of a discrete-time coalescent with simultaneous multiple mergers of ancestral lineages. The results extend those obtained by Huillet and the author in `Population genetics models with skewed fertilities: a forward and backward analysis', Stochastic Models 27 (2011), 521-554.

In this note we study a class of $2\times 2$ Polya-Eggenberger urn models, which serves as a stochastic model in biology describing cannibalistic behavior of populations. A special case was studied before by Pittel using asymptotic approximation techniques, and more recently by Hwang et al. using generating functions. We obtain limit laws for the stated class of so-called cannibal urns by using Pittel's method, and also different techniques.

The objective of this paper is to establish explicit concentration inequalities for the Glauber dynamics related with continuum or discrete Gibbs measures. At first we establish the optimal transportation-information $W_1 I$-inequality for the $M/M/\infty$-queue associated with the Poisson measure, which improves several previous known results. Under the Dobrushin's uniqueness condition, we obtain some explicit $W_1 I$-inequalities for Gibbs measures both in the continuum and in the discrete lattice. Our method is a combination of Lipschitzian spectral gap, the Lyapunov test function approach and the tensorization technique.

In the paper [7], Guasoni studies financial markets which are subject to proportional transaction costs. The standard martingale framework of stochastic finance is not applicable in these markets, since the transaction costs force trading strategies to have bounded variation, while continuous- time martingale strategies have infinite transaction cost. The main question that arises out of [7] is whether it is possible to give a convenient condition to guarantee that a trading strategy has no arbitrage. Such a condition was proposed and studied in [6] and [1], the so-called stickiness property, whereby an asset's price is never certain to exit a ball within a predetermined finite time. In this paper, we define the multidimensional extension of the stickiness property, to handle arbitrage-free conditions for markets with multiple assets and proportional transaction costs. We show that this condition is sufficient for a multi-asset model to be free of arbitrage. We also show that d-dimensional fractional Brownian models are jointly sticky, and we establish a time-change result for joint stickiness.

Revisiting an article by Chernin and Ibragimov on unimodality of stable laws, we show that their approach to deduce the general case from the extremal ones, whose completion contained an error as discovered later by Kanter, can be carried out successfully in considering Bochner's subordination and multiplicative strong unimodality. This short proof of the unimodality of all stable densities yields also a multiplicative counterpart to Yamazato's additive ones.

We prove that the probability $P_N$ for a real random Gaussian $N\times N\times 2$ tensor to be of real rank $N$ is $P_N=(\Gamma((N+1)/2))^N/G(N+1)$, where $\Gamma(x)$, $G(x)$ denote the gamma and Barnes $G$-functions respectively. This is a rational number for $N$ odd and a rational number multiplied by $\pi^{N/2}$ for $N$ even. The probability to be of rank $N+1$ is $1-P_N$. The proof makes use of recent results on the probability of having $k$ real generalized eigenvalues for real random Gaussian $N\times N$ matrices. We also prove that $\log P_N= (N^2/4)\log (e/4)+(\log N-1)/12-\zeta '(-1)+{\rm O}(1/N)$ for large $N$, where $\zeta$ is the Riemann zeta function.

We give characterisations for Brownian motion and continuous local martingales, using the crossing tree, which is a sample-path decomposition based on first-passages at nested scales. These results are based on ideas used in the construction of Brownian motion on the Sierpinski gasket (Barlow and Perkins 1988). Using our characterisation we propose a test for the continuous martingale hypothesis, that is, that a given process is a continuous local martingale. The crossing tree gives a natural break-down of a sample path at different spatial scales, which we use to investigate the scale at which a process looks like a continuous local martingale. Simulation experiments indicate that our test is more powerful than an alternative approach which uses the sample quadratic variation.

In this note, we explore applications of a known lemma which relates the expected exit time of Brownian motion from a simply connected domain with the power series of a conformal map into that domain. We use the lemma to calculate the expected exit time from a number of domains, and in the process describe a probabilistic method for summing certain series. In particular, we give a proof of Euler's classical result that $\zeta(2) = \pi^2/6$. We also show how the relationship between the power series and the Brownian exit time gives several immediate consequences when teamed with a deep result of de Branges concerning the coefficients of power series of normalized conformal maps. We conclude by stating an extension of the lemma in question to domains which are not simply connected.

We study questions of existence and uniqueness of weak and strong solutions for a one-sided Tanaka equation with constant drift lambda. We observe a dichotomy in terms of the values of the drift parameter: for $\lambda\leq 0$, there exists a strong solution which is pathwise unique, thus also unique in distribution; whereas for $\lambda > 0$, the equation has a unique in distribution weak solution, but no strong solution (and not even a weak solution that spends zero time at the origin). We also show that strength and pathwise uniqueness are restored to the equation via suitable ``Brownian perturbations".

In this note we prove the well-posedness for stochastic $2D$ Navier-Stokes equation driven by general Lévy processes (in particular, $\alpha$-stable processes), and obtain the existence of invariant measures.

Let $(e^{tA})_{t\geq0}$ be a $C_0$-contraction semigroup on a $2$-smooth Banach space $E$, let $(W_t)_{t\geq0}$ be a cylindrical Brownian motion in a Hilbert space $H$, and let $(g_t)_{t\geq0}$ be a progressively measurable process with values in the space $\gamma(H,E)$ of all $\gamma$-radonifying operators from $H$ to $E$. We prove that for all $0<p<\infty$ there exists a constant $C$, depending only on $p$and $E$, such that for all $T\geq0$ we have $$E\sup_{0\leq t\leq T}\left\Vert\int_0^t\!e^{(t-s)A}\,g_sdW_s\right\Vert^p\leq CE\left(\int_0^T\!\left(\left\Vert g_t\right\Vert_{\gamma(H,E)}\right)^2\,dt\right)^{p/2}.$$ For $p\geq2$ the proof is based on the observation that $\psi(x)=\Vert x\Vert^p$ is Fréchet differentiable and its derivative satisfies the Lipschitz estimate $\Vert \psi'(x)-\psi'(y)\Vert\leq C\left(\Vert x\Vert+\Vert y\Vert\right)^{p-2}\Vert x-y\Vert$; the extension to $0<p<2$ proceeds via Lenglart’s inequality.

We show that the limiting eigenvalue distribution of random symmetric Toeplitz matrices is absolutely continuous with density bounded by 8, partially answering a question of Bryc, Dembo and Jiang (2006). The main tool used in the proof is a spectral averaging technique from the theory of random Schrödinger operators. The similar question for Hankel matrices remains open

Abstract. Let $H(t,x)$ be the Hopf-Cole solution at time t of the Kardar-Parisi-Zhang (KPZ) equation starting with narrow wedge initial condition, i.e. the logarithm of the solution of the multiplicative stochastic heat equation starting from a Dirac delta. Also let $H^{eq}(t,x)$ be the solution at time $t$ of the KPZ equation with the same noise, but with initial condition given by a standard two-sided Brownian motion, so that $H^{eq}(t,x)-H^{eq}$ is locally of finite variation. Using the same ideas we also show that if the KPZ equation is started with a two-sided Brownian motion plus a Lipschitz function then the solution stays in this class for all time.

We give a new proof for a Ray-Knight representation of Feller's branching diffusion with logistic growth in terms of the local times of a reflected Brownian motion $H$ with a drift that is affine linear in the local time accumulated by $H$ at its current level. In Le et al. (2011) such a representation was obtained by an approximation through Harris paths that code the genealogies of particle systems. The present proof is purely in terms of stochastic analysis, and is inspired by previous work of Norris, Rogers and Williams (1988).

Regular $g$-measures are discrete-time processes determined by conditional expectations with respect to the past. One-dimensional Gibbs measures, on the other hand, are fields determined by simultaneous conditioning on past and future. For the Markovian and exponentially continuous cases both theories are known to be equivalent. Its equivalence for more general cases was an open problem. We present a simple example settling this issue in a negative way: there exist $g$-measures that are continuous and non-null but are not Gibbsian. Our example belongs, in fact, to a well-studied family of processes with rather nice attributes: It is a chain with variable-length memory, characterized by the absence of phase coexistence and the existence of a visible renewal scheme

Let $\Lambda$ be the limiting smallest eigenvalue in the general $(\beta,a)$-Laguerre ensemble of random matrix theory. That is, $\Lambda$ is the $n\to\infty$ distributional limit of the (scaled) minimal point drawn from the density proportional to $\Pi_1\leq i\leq j\leq n$ $$\left|\lambda_i-\lambda_j\right|^\beta\prod_{i=1}^n\lambda_i^{\frac{\beta}{2}(a+1)-1}e^{-\frac{\beta}{2}\lambda_i}$$ on $(\mathbb{R}_+^n$. Here $\beta>0$, $a> -1$; for $\beta=1,2,4$ and integer $a$, this object governs the singular values of certain rank $n$ Gaussian matrices. We prove that $$ \mathbb{P}(\Lambda>\lambda)=e^{-\frac{\beta}{2}\lambda+2\gamma\sqrt{\lambda}}\lambda^{-\frac{\gamma(\gamma+1-\beta/2)}{2\beta}} e(\beta,a)(1+o(1))$$ as $\lambda\to\infty$ in which $$\gamma = \frac{\beta}{2} (a+1)-1$$ and $e(\beta, a) > 0$ is a constant (which we do not determine). This estimate complements/extends various results previously available for special values of $\beta$ and $a$.

We prove correlation inequalities for linearly edge-reinforced random walk. These correlation inequalities concern the first entry tree, i.e. the tree of edges used to enter any vertex for the first time. They also involve the asymptotic fraction of time spent on particular edges. Basic ingredients are known FKG-type inequalities and known negative associations for determinantal processes.

In this note, we provide a simulation algorithm for a diffusion process in a layered media. Our main tools are the properties of the Skew Brownian motion and a path decomposition technique for simulating occupation times.

This note is motivated by results of Angel, Goodman, den Hollander and Slade (2008) and Da, Sapozhnikov and Vagvolgyi (2009) about global relations between the invasion percolation cluster (IPC) and the incipient infinite cluster (IIC) on regular trees and on two dimensional lattices, respectively. Namely, that the laws of the two objects are mutually singular, and, in the case of regular trees, that the IIC stochastically dominates the IPC. We prove that on two dimensional lattices, the IIC does not stochastically dominate the IPC. This is the first example showing that the relation between the IIC and IPC is significantly different on trees and in two dimensions.

Supercritical branching processes in constant environment conditioned on eventual extinction are known to be subcritical branching processes. The case of random environment is more subtle. A supercritical branching diffusion in random environment (BDRE) conditioned on eventual extinction of the population is not a branching diffusion in a homogeneous environment. However the law of the population size of a supercritical BDRE (averaged over the environment) conditioned on eventual extinction is equal to the law of the population size of a subcritical BDRE (averaged over the environment). As a consequence, supercritical BDREs have a phase transition which is similar to a well-known phase transition of subcritical branching processes in random environment.

We consider the voter model on $\mathbb{Z}$, starting with all 1's to the left of the origin and all $0$'s to the right of the origin. It is known that if the associated random walk kernel $p$ has zero mean and a finite r-th moment for any $r>3$, then the evolution of the boundaries of the interface region between 1's and 0's converge in distribution to a standard Brownian motion $(B_t)_{t>0}$ under diffusive scaling of space and time. This convergence fails when $p$ has an infinite $r$-th moment for any $r<3$, due to the loss of tightness caused by a few isolated $1$'s appearing deep within the regions of all $0$'s (and vice versa) at exceptional times. In this note, we show that as long as $p$ has a finite second moment, the measure-valued process induced by the rescaled voter model configuration is tight, and converges weakly to the measure-valued process $1_{x< B_t} dx$, $t>0$.