Joint distribution of the process and its sojourn time for pseudo-processes governed by high-order heat equation

Consider the high-order heat-type equation $\partial u/\partial t=\pm \partial^N u/\partial x^N$ for an integer $N>2$ and introduce the related Markov pseudo-process $(X(t))_{t\ge 0}$. In this paper, we study the sojourn time $T(t)$ in the interval $[0,+\infty)$ up to a fixed time $t$ for this pseudo-process. We provide explicit expressions for the joint distribution of the couple $(T(t),X(t))$

On the most visited sites of planar Brownian motion

Let (B_t : t > 0) be a planar Brownian motion and define gauge functions $\phi_\alpha(s)=log(1/s)^{-\alpha}$ for $\alpha>0$. If $\alpha<1$ we show that almost surely there exists a point x in the plane such that $H^{\phi_\alpha}({t > 0 : B_t=x})>0$, but if $\alpha>1$ almost surely $H^{\phi_\alpha} ({t > 0 : B_t=x})=0$ simultaneously for all $x\in R^2$. This resolves a longstanding open problem posed by S.,J. Taylor in 1986

Fluctuations of the Total Number of Critical Points of Random Spherical Harmonics

We determine the asymptotic law for the fluctuations of the total number of critical points of random Gaussian spherical harmonics in the high degree limit. Our results have implications on the sophistication degree of an appropriate percolation process for modelling nodal domains of eigenfunctions on generic compact surfaces or billiards

Joint distribution of the process and its sojourn time on the positive half-line for pseudo-processes governed by high-order heat equation.

Consider the high-order heat-type equation ∂u/∂t = ±∂Nu/∂xN for an integer N > 2 and introduce the related Markov pseudo-process (X(t))t≥0. In this paper, we study the sojourn time T(t) in the interval [0, +∞) up to a fixed time t for this pseudo-process. We provide explicit expressions for the joint distribution of the couple (T(t),X(t))

On the Correlation of Critical Points and Angular Trispectrum for Random Spherical Harmonics

We prove a Central Limit Theorem for the Critical Points of Random Spherical Harmonics, in the High-Energy Limit. The result is a consequence of a deeper characterizations of the total number of critical points, which are shown to be asymptotically fully correlated with the sample trispectrum, i.e., the integral of the fourth Hermite polynomial evaluated on the eigenfunctions themselves. As a consequence, the total number of critical points and the nodal length are fully correlated for random spherical harmonics, in the high-energy limit

Entrance and sojourn times for Markov chains. Application to (L,R)(L,R)-random walks

In this paper, we provide a methodology for computing the probability distribution of sojourn times for a wide class of Markov chains. Our methodology consists in writing out linear systems and matrix equations for generating functions involving relations with entrance times. We apply the developed methodology to some classes of random walks with bounded integer-valued jumps.Comment: 30 page

A quantitative central limit theorem for the Euler–Poincaré characteristic of random spherical eigenfunctions.

We establish here a quantitative central limit theorem (in Wasserstein distance) for the Euler–Poincaré characteristic of excursion sets of random spherical eigenfunctions in dimension 2. Our proof is based upon a decomposition of the Euler–Poincaré characteristic into different Wiener-chaos components: we prove that its asymptotic behaviour is dominated by a single term, corresponding to the chaotic component of order two. As a consequence, we show how the asymptotic dependence on the threshold level u is fully degenerate, that is, the Euler–Poincaré characteristic converges to a single random variable times a deterministic function of the threshold. This deterministic function has a zero at the origin, where the variance is thus asymptotically of smaller order. We discuss also a possible unifying framework for the Lipschitz–Killing curvatures of the excursion sets for Gaussian spherical harmonics

Joint distribution of the process and its sojourn time in a half-line [a,+∞)[a,+\infty) for pseudo-processes driven by a high-order heat-type equation.

Let (X(t))t≥0 be the pseudo-process driven by the high-order heat-type equation ∂u = ± ∂Nu , ∂t ∂xN where N is an integer greater than 2. We consider the sojourn time spent by (X(t))t≥0 in [a,+∞) (a ∈ R), up to a fixed time t > 0: Ta(t) = 0t 1l[a,+∞)(X(s)) ds. The purpose of this paper is to explicit the joint pseudo-distribution of the vector (Ta(t),X(t)) when the pseudo-process starts at a point x ∈ R at time 0. The method consists in solving a boundary value problem satisfied by the Laplace transform of the aforementioned distribution