Showing papers in "Electronic Communications in Probability in 2016"
TL;DR: In this paper, the authors considered the smoothed multiplicative noise stochastic heat equation and showed that the solution exhibits weak disorder when β > 0, where β is a parameter.
Abstract: We consider the smoothed multiplicative noise stochastic heat equation \[\mathrm{d} u_{\varepsilon ,t}= \frac 12 \Delta u_{\varepsilon ,t} \mathrm{d} t+ \beta \varepsilon ^{\frac{d-2} {2}}\, \, u_{\varepsilon , t} \, \mathrm{d} B_{\varepsilon ,t} , \;\;u_{\varepsilon ,0}=1,\] in dimension $d\geq 3$, where $B_{\varepsilon ,t}$ is a spatially smoothed (at scale $\varepsilon $) space-time white noise, and $\beta >0$ is a parameter. We show the existence of a $\bar \beta \in (0,\infty )$ so that the solution exhibits weak disorder when $\beta \bar \beta $. The proof techniques use elements of the theory of the Gaussian multiplicative chaos.
46 citations
TL;DR: In this paper, the authors established uniform bounds on the low-order derivatives of Stein equation solutions for a broad class of multivariate, strongly log-concave target distributions.
Abstract: We establish uniform bounds on the low-order derivatives of Stein equation solutions for a broad class of multivariate, strongly log-concave target distributions. These “Stein factor” bounds deliver control over Wasserstein and related smooth function distances and are well-suited to analyzing the computable Stein discrepancy measures of Gorham and Mackey. Our arguments of proof are probabilistic and feature the synchronous coupling of multiple overdamped Langevin diffusions.
36 citations
TL;DR: In this article, the existence of a random Lipschitz function F : Zd−1 → Z+ such that, for every x ∈ Zd −1, the site (x, F(x)) is open in a site percolation process on Zd.
Abstract: We prove the existence of a (random) Lipschitz function F : Zd−1 → Z+ such that, for every x ∈ Zd−1, the site (x, F(x)) is open in a site percolation process on Zd . The Lipschitz constant may be taken to be 1 when the parameter p of the percolation model is sufficiently close to 1. Alexander Holroyd
32 citations
TL;DR: In this paper, a generalized space-fractional Poisson process is proposed to preserve the characterizing Levy property, which is related to Prabhakar derivatives, specific convolution-type integral operators.
Abstract: The space-fractional Poisson process is a time-changed homogeneous Poisson process where the time change is an independent stable subordinator. In this paper, a further generalization is discussed that preserves the Levy property. We introduce a generalized process by suitably time-changing a superposition of weighted space-fractional Poisson processes. This generalized process can be related to a specific subordinator
for which it is possible to explicitly write the characterizing Levy measure. Connections are highlighted to Prabhakar derivatives, specific convolution-type integral operators. Finally, we study the effect of introducing Prabhakar derivatives also in time.
31 citations
TL;DR: In particular, this paper showed that when one superimposes the random current model related to the Ising model with an independent Bernoulli percolation model with well-chosen weights, one obtains exactly the FK-percolation (or random cluster model) associated with the ising model, and this relation can be interpreted via loop-soups.
Abstract: We make a few elementary observations that relate directly the items mentioned in the title. In particular, we note that when one superimposes the random current model related to the Ising model with an independent Bernoulli percolation model with well-chosen weights, one obtains exactly the FK-percolation (or random cluster model) associated with the Ising model, and we point out that this relation can be interpreted via loop-soups, combining the description of the sign of a Gaussian free field on a discrete graph knowing its square (and the relation of this question with the FK-Ising model) with the loop-soup interpretation of the random current model.
25 citations
TL;DR: In this article, the authors studied the binary branching Brownian motion with spatially-inhomogeneous branching rate and showed that the distribution of the rightmost particle centred about the Dirac delta function converges to a mixture of Gumbel distributions according to a martingale limit.
Abstract: We study the model of binary branching Brownian motion with spatially-inhomogeneous branching rate $\beta \delta _0(\cdot )$, where $\delta _0(\cdot )$ is the Dirac delta function and $\beta $ is some positive constant. We show that the distribution of the rightmost particle centred about $\frac{\beta } {2}t$ converges to a mixture of Gumbel distributions according to a martingale limit. Our results form a natural extension to S. Lalley and T. Sellke [10] for the degenerate case of catalytic branching.
25 citations
TL;DR: In this paper, it was shown that the frog model on the rooted $d$-ary tree changes from transient to recurrent as the number of frogs per site is increased, and that the location of this transition is on the same order as the degree of the tree.
Abstract: The frog model on the rooted $d$-ary tree changes from transient to recurrent as the number of frogs per site is increased. We prove that the location of this transition is on the same order as the degree of the tree.
21 citations
TL;DR: Among all generalized Ornstein-Uhlenbeck processes which sample the same invariant measure and for which the same amount of randomness (a $N$-dimensional Brownian motion) is injected in the system, the asymptotic rate of convergence is maximized by a non-reversible hypoelliptic one as discussed by the authors.
Abstract: Among all generalized Ornstein-Uhlenbeck processes which sample the same invariant measure and for which the same amount of randomness (a $N$-dimensional Brownian motion) is injected in the system, we prove that the asymptotic rate of convergence is maximized by a non-reversible hypoelliptic one.
21 citations
TL;DR: In this paper, a version of the multidimensional Fourth Moment Theorem for chaotic random vectors, in the general context of diffusion Markov generators, was shown to imply joint convergence of a given sequence of vectors.
Abstract: We prove a version of the multidimensional Fourth Moment Theorem for chaotic random vectors, in the general context of diffusion Markov generators. In addition to the usual componentwise convergence and unlike the infinite-dimensional Ornstein-Uhlenbeck generator case, another moment-type condition is required to imply joint convergence of of a given sequence of vectors.
19 citations
TL;DR: A strong pathwise Siegmund dual is identified of the ancestor in a two-type Wright-Fisher population with mutation and selection, conditional on the overall type frequency in the old population, and the equilibrium tail probabilities of $L$ are characterised in terms of hitting probabilities of the dual process.
Abstract: Using graphical methods based on a ‘lookdown’ and pruned version of the ancestral selection graph, we obtain a representation of the type distribution of the ancestor in a two-type Wright-Fisher population with mutation and selection, conditional on the overall type frequency in the old population. This extends results from [17] to the case of heavy-tailed offspring, directed by a reproduction measure $\Lambda$. The representation is in terms of the equilibrium tail probabilities of the line-counting process $L$ of the graph. We identify a strong pathwise Siegmund dual of $L$, and characterise the equilibrium tail probabilities of $L$ in terms of hitting probabilities of the dual process.
19 citations
TL;DR: In this article, the authors studied the closed convex hull of the random set Zs of a Levy process, and provided conditions for the integration of the intrinsic volumes of Zs and for their means in the case of symmetric α-stable Levy processes.
Abstract: Let X(t), t ≥ 0, be a Levy process in Rd starting at the origin. We study the closed
convex hull Zs of {X(t) : 0 ≤ t ≤ s}. In particular, we provide conditions for the
integrability of the intrinsic volumes of the random set Zs and find explicit expressions for their means in the case of symmetric α-stable Levy processes. If the process is symmetric and each its one-dimensional projection is non-atomic, we establish that the origin a.s. belongs to the interior of Zs for all s > 0. Limit theorems for the convex hull of Levy processes with normal and stable limits are also obtained.
TL;DR: In this article, the authors generalize the exactly solvable corner growth models by choosing the rate of the exponential distribution and the parameter of the geometric distribution at site $(i, j)$, where $(a_i)_{i \ge 1}$ and $(b_j)_{j \ge 2} are jointly ergodic random sequences.
Abstract: We generalize the exactly solvable corner growth models by choosing the rate of the exponential distribution $a_i+b_j$ and the parameter of the geometric distribution $a_i b_j$ at site $(i, j)$, where $(a_i)_{i \ge 1}$ and $(b_j)_{j \ge 1}$ are jointly ergodic random sequences. We identify the shape function in terms of a simple variational problem, which can be solved explicitly in some special cases.
TL;DR: In this paper, the Kesten-Grincevicius-Goldie theorem holds for the perpetuity equation with respect to a nonconstant slowly varying function, where the renewal theoretic approach is used.
Abstract: Consider the perpetuity equation $X \stackrel{\mathcal {D}} {=} A X + B$, where $(A,B)$ and $X$ on the right-hand side are independent. The Kesten–Grincevicius–Goldie theorem states that if $\mathbf{E} A^\kappa = 1$, $\mathbf{E} A^\kappa \log _+ A x \} \sim c x^{-\kappa }$. Assume that $\mathbf{E} |B|^
u \kappa $, and consider two cases (i) $\mathbf{E} A^\kappa = 1$, $\mathbf{E} A^\kappa \log _+ A = \infty $; (ii) $\mathbf{E} A^\kappa \kappa $. We show that under appropriate additional assumptions on $A$ the asymptotic $\mathbf{P} \{ X > x \} \sim c x^{-\kappa } \ell (x) $ holds, where $\ell $ is a nonconstant slowly varying function. We use Goldie’s renewal theoretic approach.
TL;DR: In this paper, the mutual regularity properties of Palm measures of point processes are studied, and it is shown that a key determining factor for these properties is the rigidity behaviour exhibited by the point process in question.
Abstract: We study the mutual regularity properties of Palm measures of point processes, and establish that a key determining factor for these properties is the rigidity behaviour that is exhibited by the point process in question. Thereby, we extend the results of [OsSh] to new ensembles, including the zeroes of the standard planar Gaussian analytic function and several others.
TL;DR: In this paper, the authors prove the occurrence of a phase transition accompanied by the emergence of cycles of diverging lengths in the random interchange process on the hypercube and prove the existence of a cycle-cycle transition.
Abstract: We prove the occurrence of a phase transition accompanied by the emergence of cycles of diverging lengths in the random interchange process on the hypercube.
TL;DR: In this article, a q-deformed version of the uniform Gibbs measure on dimers on the periodized hexagonal lattice was considered and it was shown that it is invariant under a certain irreversible q-Whittaker dynamic.
Abstract: We consider a q-deformed version of the uniform Gibbs measure on dimers on the periodized hexagonal lattice (equivalently, on interlacing particle configurations, if vertical dimers are seen as particles) and show that it is invariant under a certain irreversible q-Whittaker dynamic. Thereby we provide a new non-trivial example of driven interacting two-dimensional particle system, or of (2+1)-dimensional stochastic growth model, with explicit stationary measure. We emphasize that this measure is far from being a product Bernoulli measure. These Gibbs measures and dynamics both arose earlier in the theory of Macdonald processes. The q=0 degeneration of the Gibbs measures reduce to the usual uniform dimer measures with given tilt, the degeneration of the dynamics originate in the study of Schur processes and the degeneration of the results contained herein were recently treated in work of the second author.
TL;DR: In this paper, necessary and sufficient conditions for the Marchenko-Pastur theorem for matrices with IID isotropic rows were obtained for certain quadratic forms of the rows.
Abstract: We obtain necessary and sufficient conditions for the Marchenko-Pastur theorem for matrices with IID isotropic rows. Our conditions are related to a weak concentration property for certain quadratic forms of the rows.
TL;DR: In this article, a one-dimensional, continuous, regular, and strong Markov process with state space (E$) is shown to hit any point in E$ with positive probability.
Abstract: A one-dimensional, continuous, regular, and strong Markov process $X$ with state space $E$ hits any point $z \in E$ fast with positive probability. To wit, if ${\boldsymbol{\tau } }_z = \inf \{t \geq 0:X_{t} = z\}$, then $\textsf{P} _\xi ({\boldsymbol{\tau } }_z 0$ for all $\xi \in E$ and $\varepsilon >0$.
TL;DR: The nature of the alignment with gaps corresponding to a Longest Common Subsequence of two independent iid random sequences drawn from a finite alphabet is investigated and it is shown that such an optimal alignment typically matches pieces of similar length.
Abstract: The nature of the alignment with gaps corresponding to a longest common subsequence (LCS) of two independent iid random sequences drawn from a finite alphabet is investigated. It is shown that such an optimal alignment typically matches pieces of similar short-length. This is of importance in understanding the structure of optimal alignments of two sequences. Moreover, it is also shown that any property, common to two subsequences, typically holds in most parts of the optimal alignment whenever this same property holds, with high probability, for strings of similar short-length. Our results should, in particular, prove useful for simulations since they imply that the re-scaled two dimensional representation of a LCS gets uniformly close to the diagonal as the length of the sequences grows without bound.
TL;DR: In this paper, the authors consider time-dependent random walks among timedependent conductances and show that two-sided Gaussian heat kernel estimates are not stable under perturbations.
Abstract: We consider time-dependent random walks among time-dependent conductances. For discrete time random walks, we show that, unlike the time-independent case, two-sided Gaussian heat kernel estimates are not stable under perturbations. This is proved by giving an example of a ballistic and transient time-dependent random walk on $\mathbb{Z}$ among uniformly elliptic time-dependent conductances. For continuous time random walks, we show the instability when the holding times are i.i.d. $\exp(1)$, and in contrast, we prove the stability when the holding times change by sites in such a way that the base measure is a uniform measure.
TL;DR: In this paper, the authors provided a new proof of a Kramers' type law for self-stabilizing diffusion, which correspond to the hydrodynamical limit of a mean-field system of particles and may be seen as the probabilistic interpretation of the granular media equation.
Abstract: We provide a new proof of a Kramers’ type law for self-stabilizing diffusion. These diffusions correspond to the hydrodynamical limit of a mean-field system of particles and may be seen as the probabilistic interpretation of the granular media equation. We use the same hypotheses as the ones used in the work “Large deviations and a Kramers’ type law for self-stabilizing diffusions” by Herrmann, Imkeller and Peithmann in which the authors obtain a first proof of the statement.
TL;DR: In this paper, it was shown that if the additive martingale condition is not verified, then the growth-fragmentation process explodes almost surely, and the authors used this condition to obtain a spine decomposition of the process.
Abstract: Markovian growth-fragmentation processes describe a family of particles which can grow larger or smaller with time, and occasionally split in a conservative manner. They were introduced in [3], where special attention was given to the self-similar case. A Malthusian condition was notably given under which the process does not locally explode, in the sense that for all times, the masses of all the particles can be listed in non-increasing order. Our main result in this work states the converse: when this condition is not verified, then the growth-fragmentation process explodes almost surely. Our proof involves using the additive martingale to bias the probability measure and obtain a spine decomposition of the process, as well as properties of self-similar Markov processes.
TL;DR: In this paper, a non-Markovian optimal stopping problem on finite horizon is considered and the value process can be represented by means of a backward stochastic differential equation (BSDE), defined on an enlarged probability space, containing a one-jump point process as integrator and an unknown process with a sign constraint as integrand.
Abstract: We consider a non-Markovian optimal stopping problem on finite horizon. We prove that the value process can be represented by means of a backward stochastic differential equation (BSDE), defined on an enlarged probability space, containing a stochastic integral having a one-jump point process as integrator and an (unknown) process with a sign constraint as integrand. This provides an alternative representation with respect to the classical one given by a reflected BSDE. The connection between the two BSDEs is also clarified. Finally, we prove that the value of the optimal stopping problem is the same as the value of an auxiliary optimization problem where the intensity of the point process is controlled.
TL;DR: In this article, the authors consider the random walk loop soup on the discrete half-plane and study the percolation problem, i.e. the existence of an infinite cluster of loops.
Abstract: We consider the random walk loop soup on the discrete half-plane $\mathbb{Z} \times \mathbb{N} ^{\ast }$ and study the percolation problem, i.e. the existence of an infinite cluster of loops. We show that the critical value of the intensity is equal to $\frac{1} {2}$. The absence of percolation at intensity $\frac{1} {2}$ was shown in a previous work. We also show that in the supercritical regime, one can keep only the loops up to some large enough upper bound on the diameter and still have percolation.
TL;DR: The variational principle for Gibbs point processes with general finite range interaction was proved in this paper, where the Gibbs point process is identified as the minimizers of the free excess energy equals to the sum of the specific entropy and the mean energy.
Abstract: The variational principle for Gibbs point processes with general finite range interaction is proved. Namely, the Gibbs point processes are identified as the minimizers of the free excess energy equals to the sum of the specific entropy and the mean energy. The interaction is very general and includes superstable pairwise potential, finite or infinite multibody potential, geometrical interaction, hardcore interaction. The only restrictive assumption involves the finite range property.
TL;DR: In this article, the authors explore a set of probabilistic models related to the SABR model in mathematical finance, which can be obtained by geometry-preserving transformations, and show how to translate the properties of the hyperbolic Brownian motion (density, probability mass, drift) to each particular model.
Abstract: Starting from the hyperbolic Brownian motion as a time-changed Brownian motion, we explore a set of probabilistic models–related to the SABR model in mathematical finance–which can be obtained by geometry-preserving transformations, and show how to translate the properties of the hyperbolic Brownian motion (density, probability mass, drift) to each particular model. Our main result is an explicit expression for the probability of any of these models hitting the boundary of their domains, the proof of which relies on the properties of the aforementioned transformations as well as time-change methods.
TL;DR: In this paper, the authors proved almost-sure convergence for the self-attracting diffusion on the unit sphere, and showed that the self attracting diffusion is almost sure to converge.
Abstract: This paper proves almost-sure convergence for the self attracting diffusion on the unit sphere \[ dX_t=
u \circ dW_{t}(X_t)-a\int _{0}^{t}
abla _{\mathbb{S} ^n}V_{X_s}(X_t) dsdt,\qquad X_0=x\in \mathbb{S} ^n, \] where $
u >0$, $a 0$.
TL;DR: In this paper, it was shown that matrix elements of functions of Wigner matrices fluctuate on a scale of order O(n − 1/2 ) and that the limiting fluctuation is a function of the matrix that has bounded variation.
Abstract: We show that matrix elements of functions of $N\times N$ Wigner matrices fluctuate on a scale of order $N^{-1/2}$ and we identify the limiting fluctuation Our result holds for any function $f$ of the matrix that has bounded variation thus considerably relaxing the regularity requirement imposed in [7, 11]
TL;DR: In this article, the M1 topology is defined for cadlag paths taking values in the space of tempered distributions in the dual of a countably Hilbertian nuclear space, and Compactness and tightness characterisations are derived which allow us to study a collection of stochastic processes through their projections on the familiar space of real-valued cadlag processes.
Abstract: Skorokhod’s M1 topology is defined for cadlag paths taking values in the space of tempered distributions (more generally, in the dual of a countably Hilbertian nuclear space). Compactness and tightness characterisations are derived which allow us to study a collection of stochastic processes through their projections on the familiar space of real-valued cadlag processes. It is shown how this topological space can be used in analysing the convergence of empirical process approximations to distribution-valued evolution equations with Dirichlet boundary conditions.
TL;DR: In this article, it was proved by Hoeffding in 1963 that a real random variable $X$ confined to $[a,b]$ satisfies the following property:
Abstract: It was proved by Hoeffding in 1963 that a real random variable $X$ confined to $[a,b]$ satisfies $\mathbb{E} \, e^{X-\operatorname{\mathbb {E}} X} \le e^{(b-a)^2/8}$. We generalise this to complex random variables.