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On the stationary distribution of reflected Brownian motion in a wedge: differential properties
Bousquet-Mélou M., Elvey Price A., Franceschi S., Hardouin C., Raschel K.
Q1
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Recurrence and transience of the critical random walk snake in random conductances
Legrand A., Sabot C., Schapira B.
Q1
Noise-like analytic properties of imaginary chaos
Aru J., Baverez G., Jego A., Junnila J.
Q1
Quantitative hydrodynamic limits of the Langevin dynamics for gradient interface models
Armstrong S., Dario P.
Q1
A Palm space approach to non-linear Hawkes processes
Robert P., Vignoud G.
Q1
Law of large numbers for the maximum of the two-dimensional Coulomb gas potential
Lambert G., Leblé T., Zeitouni O.
Q1
Hydrodynamic limit of the Schelling model with spontaneous Glauber and Kawasaki dynamics
Barret F., Torri N.
Q1
A phase transition in block-weighted random maps
Fleurat W., Salvy Z.
Q1
A branching process with coalescence to model random phylogenetic networks
Bienvenu F., Duchamps J.
Q1
Asymptotic normality of pattern counts in conjugacy classes
Féray V., Kammoun M.S.
Q1
Uniform-in-time propagation of chaos for kinetic mean field Langevin dynamics
Chen F., Lin Y., Ren Z., Wang S.
Q1
Concentration estimates for slowly time-dependent singular SPDEs on the two-dimensional torus
Berglund N., Nader R.
Q1
General criteria for the study of quasi-stationarity
Champagnat N., Villemonais D.
Q1
For Markov processes with absorption, we provide general criteria ensuring the existence and the exponential non-uniform convergence in weighted total variation norm to a quasi-stationary distribution. We also characterize a subset of its domain of attraction by an integrability condition, prove the existence of a right eigenvector for the semigroup of the process and the existence and exponential ergodicity of the Q-process. These results are applied to one-dimensional and multi-dimensional diffusion processes, to pure jump continuous time processes, to reducible processes with several communication classes, to perturbed dynamical systems and discrete time processes evolving in discrete state spaces.
Scaling limit of linearly edge-reinforced random walks on critical Galton-Watson trees
Andriopoulos G., Archer E.
Q1
We prove an invariance principle for linearly edge reinforced random walks on γ-stable critical Galton-Watson trees, where γ∈(1,2] and where the edge joining x to its parent has rescaled initial weight d(O,x)α for some α≤1. This corresponds to the recurrent regime of initial weights. We then establish fine asymptotics for the limit process. In the transient regime, we also give an upper bound on the random walk displacement in the discrete setting, showing that the edge reinforced random walk never has positive speed, even when the initial edge weights are strongly biased away from the root.
On the Besicovitch-stability of noisy random tilings
Léo G., Mathieu S.
Q1
In this paper, we introduce a noisy framework for SFTs, allowing some amount of forbidden patterns to appear. Using the Besicovitch distance, which permits a global comparison of configurations, we then study the closeness of noisy measures to non-noisy ones as the amount of noise goes to 0. Our first main result is the full classification of the (in)stability in the one-dimensional case. Our second main result is a stability property under Bernoulli noise for higher-dimensional periodic SFTs, which we finally extend to an aperiodic example through a variant of the Robinson tiling.
Evolution of a passive particle in a one-dimensional diffusive environment
Huveneers F., Simenhaus F.
Q1
We study the behavior of a tracer particle driven by a one-dimensional fluctuating potential, defined initially as a Brownian motion, and evolving in time according to the heat equation. We obtain two main results. First, in the short time limit, we show that the fluctuations of the particle become Gaussian and sub-diffusive, with dynamical exponent 3∕4. Second, in the long time limit, we show that the particle is trapped by the local minima of the potential and evolves diffusively i.e. with exponent 1∕2.
Mean-field bounds for Poisson-Boolean percolation
Dewan V., Muirhead S.
Q1
We establish the mean-field bounds γ≥1, δ≥2 and △≥2 on the critical exponents of the Poisson-Boolean continuum percolation model under a moment condition on the radii; these were previously known only in the special case of fixed radii (in the case of γ), or not at all (in the case of δ and △). We deduce these as consequences of the mean-field bound β≤1, recently established under the same moment condition [8], using a relative entropy method introduced by the authors in previous work [7].
Time reversal of spinal processes for linear and non-linear branching processes near stationarity
Henry B., Méléard S., Tran V.C.
Q1
We consider a stochastic individual-based population model with competition, trait-structure affecting reproduction and survival, and changing environment. The changes of traits are described by jump processes, and the dynamics can be approximated in large population by a non-linear PDE with a non-local mutation operator. Using the fact that this PDE admits a non-trivial stationary solution, we can approximate the non-linear stochastic population process by a linear birth-death process where the interactions are frozen, as long as the population remains close to this equilibrium. This allows us to derive, when the population is large, the equation satisfied by the ancestral lineage of an individual uniformly sampled at a fixed time T, which is the path constituted of the traits of the ancestors of this individual in past times t≤T. This process is a time inhomogeneous Markov process, but we show that the time reversal of this process possesses a very simple structure (e.g. time-homogeneous and independent of T). This extends recent results where the authors studied a similar model with a Laplacian operator but where the methods essentially relied on the Gaussian nature of the mutations.
Invariant measures of critical branching random walks in high dimension
Rapenne V.
Q1
In this work, we characterize cluster-invariant point processes for critical branching spatial processes on Rd for all large enough d when the motion law is α-stable or has a finite discrete range. More precisely, when the motion is α-stable with α≤2 and the offspring law μ of the branching process has an heavy tail such that μ(k)∼k−2−β, then we need the dimension d to be strictly larger than the critical dimension α∕β. In particular, when the motion is Brownian and the offspring law μ has a second moment, this critical dimension is 2. Contrary to the previous work of Bramson, Cox and Greven in [4] whose proof used PDE techniques, our proof uses probabilistic tools only.
Sparse matrices: convergence of the characteristic polynomial seen from infinity
Coste S.
Q1
We prove that the reverse characteristic polynomial det(In−zAn) of a random n×n matrix An with iid Bernoulli(d∕n) entries converges in distribution towards the random infinite product ∏ℓ=1∞(1−zℓ)Yℓ where Yℓ are independent Poisson(dℓ∕ℓ) random variables. We show that this random function is a Poisson analog of more classical Gaussian objects such as the Gaussian holomorphic chaos. As a byproduct, we obtain new simple proofs of previous results on the asymptotic behaviour of extremal eigenvalues of sparse Erdős-Rényi digraphs: for every d>1, the greatest eigenvalue of An is close to d and the second greatest is smaller than d, a Ramanujan-like property for irregular digraphs. For d
Ramification of Volterra-type rough paths
Bruned Y., Katsetsiadis F.
Q1
We extend the new approach introduced in [24] and [25] for dealing with stochastic Volterra equations using the ideas of Rough Path theory and prove global existence and uniqueness results. The main idea of this approach is simple: Instead of the iterated integrals of a path comprising the data necessary to solve any equation driven by that path, now iterated integral convolutions with the Volterra kernel comprise said data. This leads to the corresponding abstract objects called Volterra-type Rough Paths, as well as the notion of the convolution product, an extension of the natural tensor product used in Rough Path Theory.
Creeping of Lévy processes through curves
Chaumont L., Pellas T.
Q1
