Related papers: The Liouville property and Hilbertian compression

We consider the continuous time symmetric random walk with a slow bond on $\mathbb Z$, which rates are equal to $1/2$ for all bonds, except for the bond of vertices $\{-1,0\}$, which associated rate is given by $\alpha n^{-\beta}/2$, where…

The L\'evy walk process with rests is discussed. The jumping time is governed by an $\alpha$-stable distribution with $\alpha>1$ while a waiting time distribution is Poissonian and involves a position-dependent rate which reflects a…

We obtain a Liouville property for stationary diffusions in random environment which are small, isotropic perturbations of Brownian motion in spacial dimension greater than two. Precisely, we prove that, on a subset of full probability, the…

For $\gamma \in (0,2)$, we define a weak $\gamma$-Liouville quantum gravity (LQG) metric to be a function $h\mapsto D_h$ which takes in an instance of the planar Gaussian free field (GFF) and outputs a metric on the plane satisfying a…

We have studied a random walk model based on majority rule. At a given instant, the moving direction of a cargo is determined by motor coordination mediated by a tug-of-war mechanism between two kinds of competing motor proteins. We have…

We investigate invariants for random elements of different hyperbolic groups. We provide a method, using Cayley graphs of groups, to compute the probability distribution of the minimal length of a random word, and explicitly compute the…

Let $\{\eta_{N, v}: v\in V_N\}$ be a discrete Gaussian free field in a two-dimensional box $V_N$ of side length $N$ with Dirichlet boundary conditions. We study the Liouville first passage percolation, i.e., the shortest path metric where…

We study the long-time asymptotics of the probability P_t that the Riemann-Liouville fractional Brownian motion with Hurst index H does not escape from a fixed interval [-L,L] up to time t. We show that for any H \in ]0,1], for both…

We investigate densities of vaguely continuous convolution semigroups of probability measures on $\mathbb{R}^d$. We expose that many typical conditions on the characteristic exponent repeatedly used in the literature of the subject are…

We consider a discrete time random walk in one dimension. At each time step the walker jumps by a random distance, independent from step to step, drawn from an arbitrary symmetric density function. We show that the expected positive maximum…

A theorem of L. Caffarelli implies the existence of a map pushing forward a source Gaussian measure to a target measure which is more log-concave than the source one, which contracts Euclidean distance (in fact, Caffarelli showed that the…

When the memory parameter of the elephant random walk is above a critical threshold, the process becomes superdiffusive and, once suitably normalised, converges to a non-Gaussian random variable. In a recent paper by the three first…

We study harmonic functions and Poisson boundaries for Borel probability measures on general (i.e., not necessarily locally compact) topological groups, and we prove that a second-countable topological group is amenable if and only if it…

We study the quantum mechanical Liouville model with attractive potential which is obtained by Hamiltonian symmetry reduction from the system of a free particle on $SL(2, \Real)$. The classical reduced system consists of a pair of Liouville…

We show that random walks on semisimple algebraic groups do not concentrate on proper algebraic subvarieties with uniform exponential rate of anti-concentration. This is achieved by proving a uniform spectral gap for quasi-regular…

We show that the $L^1$ norm of an exponential sum of length $X$ and with coefficients equal to the Liouville or M\"{o}bius function is at least $\gg_{\varepsilon} X^{1/4 - \varepsilon}$ for any given $\varepsilon$. For the Liouville…

The paper presents two results. The first one provides separate conditions for the upper and lower estimate of the distribution of the exit time from balls of a random walk on a weighted graph. The main result of the paper is that the lower…

We consider a weighted lattice $Z^d$ with conductance $\mu_e=|e|^{-\alpha}$. We show that the heat kernel of a variable speed random walk on it satisfies a two-sided Gaussian bound by using an intrinsic metric. We also show that when $d=2$…

The probability of a random walker to return to its starting point in dimensions one and two is unity, a theorem first proven by G. Polya. The recurrence probability -- the probability to be found at the origin at a time t, is a power law…

We consider reflecting random walks on the nonnegative integers with drift of order 1/x at height x. We establish explicit asymptotics for various probabilities associated to such walks, including the distribution of the hitting time of 0…