Related papers: Persistence exponents via perturbation theory: MA(…

The modified Bessel function of the first kind, $I_{\nu}(x)$, arises in numerous areas of study, such as physics, signal processing, probability, statistics, etc. As such, there has been much interest in recent years in deducing properties…

Algebraic persistence studies persistence modules (typically, linear representations of the poset $\mathbf{R}^n$ with $n \geq 1$) and the algebraic relationships between persistence modules that are interleaved. The notion of…

Persistence diagrams offer a way to summarize topological and geometric properties latent in datasets. While several methods have been developed that utilize persistence diagrams in statistical inference, a full Bayesian treatment remains…

We study the a.s. sample path regularity of Gaussian processes. To this end we relate the path regularity directly to the theory of small deviations. In particular, we show that if the process is $n$-times differentiable then the…

We address the problem of estimating multiple modes of a multivariate density using persistent homology, a central tool in Topological Data Analysis. We introduce a method based on the preliminary estimation of the $H_0$-persistence diagram…

We consider a reaction-diffusion process with retardation. The particles, immersed in traps initially, remain inactive until another particle is annihilated spontaneously with a rate $\lambda$ at a certain point $\vec x$. In that case the…

In this paper, we are interested in the asymptotic behaviour of the sequence of processes $(W_n(s,t))_{s,t\in[0,1]}$ with \begin{equation*} W_n(s,t):=\sum_{k=1}^{\lfloor nt\rfloor}\big(1_{\{\xi_{S_k}\leq s\}}-s\big) \end{equation*} where…

The persistence exponent, theta, is defined by N_F sim t^theta, where t is the time since the start of the coarsening process and the "no-flip fraction", N_F, is the number of points that have not seen a change of "color" since t=0. Here we…

We consider the linear stochastic recursion $x_{i+1} = a_{i}x_{i}+b_{i}$ where the multipliers $a_i$ are random and have Markovian dependence given by the exponential of a standard Brownian motion and $b_{i}$ are i.i.d. positive random…

We introduce a strategy to tackle some known obstructions of current approaches to the Fourier uniformity conjecture. Assuming GRH, we then show the conjecture holds for intervals of length at least $(\log X)^{\psi(X)}$, with $\psi(X)…

Let $E$ be a space of observables in a sequence of trials $\xi_n$ and define $m_n$ to be the empirical distributions of the outcomes. We discuss the almost sure convergence of the sequence $m_n$ in terms of the $\psi$-weak topology of…

The stability of an Auto-Regressive (AR) time sequence of finite order $L$, is determined by the maximal modulus $r^\star$ among all zeros of its generating polynomial. If $r^\star<1$ then the effect of input and initial conditions decays…

Given $N\ge2$ closed subspaces $M_1,\dotsc, M_N$ of a Hilbert space $X$, let $P_k$ denote the orthogonal projection onto $M_k$, $1\le k\le N$. It is known that the sequence $(x_n)$, defined recursively by $x_0=x$ and $x_{n+1}=P_N\cdots…

We continue our study of intermittency for the parabolic Anderson equation $\partial u/\partial t = \kappa\Delta u + \xi u$, where $u\colon \Z^d\times [0,\infty)\to\R$, $\kappa$ is the diffusion constant, $\Delta$ is the discrete Laplacian,…

Let X be some homogeneous additive functional of a skew Bessel process Y. In this note, we compute the asymptotics of the first passage time of X to some fixed level b, and study the position of Y when X exits a bounded interval [a, b]. As…

We show that the probability, P_0(l), that the height of a fluctuating (d+1)-dimensional interface in its steady state stays above its initial value up to a distance l, along any linear cut in the d-dimensional space, decays as P_0(l) \sim…

A general approach to a broad class of asymptotic problems related to long-time influence of small perturbations, of both deterministic and stochastic type, is presented in the paper. The main characteristic of this influence is a limiting…

The paper is devoted to the study of the asymptotic behaviour of Moran process in random environment, say random selection. In finite population, the Moran process may be degenerate in finite time, thus we will study its limiting process in…

We study the projection constant of the space of operators on $n$-dimensional Hilbert spaces, with the trace norm, $\mathcal S_1(n)$. We show an integral formula for the projection constant of $\mathcal S_1(n)$; namely $…

The status of different extensions of the Random Phase Approximation (RPA) is reviewed. The general framework is given within the Equation of Motion Method and the equivalent Green's function approach for the so-called Self-Consistent RPA…