arXiv subjects

Xaver Kriechbaum

Publications and source records attributed to Xaver Kriechbaum.

6 recordsLinked to original sources

arXiv2026

Multiplicative chaos from random multiplicative functions

We construct the multiplicative chaos measures emerging from the Dirichlet series of the twisted Steinhaus multiplicative function. Our work has three main features. First, our argument treats the full subcritical and critical phases simultaneously. Second, we show that the same multiplicative chaos measure arises from any approximation to the Dirichlet series in which the prime cutoff is sent to infinity and the critical line is approached from the right at arbitrary speeds. Such results are known as "universality" results, and we show this via a novel and quick partial summation trick. Third, we show that the multiplicative chaos measure is mutually absolutely continuous with a GMC measure in the same phase, coupled on the same probability space, almost surely. Our work unifies and extends results from a recent breakthrough trilogy of papers by Gorodetsky and Wong.

math.PR↗

arXiv2024

Absolute continuity of non-Gaussian and Gaussian multiplicative chaos measures

In this article, we consider the multiplicative chaos measure associated to the log-correlated random Fourier series, or random wave model, with i.i.d. coefficients taken from a general class of distributions. This measure was shown to be non-degenerate when the inverse temperature is subcritical by Junnila (Int. Math. Res. Not. 2020 (2020), no. 19, 6169-6196). When the coefficients are Gaussian, this measure is an example of a Gaussian multiplicative chaos (GMC), a well-studied universal object in the study of log-correlated fields. In the case of non-Gaussian coefficients, the resulting chaos is not a GMC in general. However, we construct a coupling between the non-Gaussian multiplicative chaos measure and a GMC such that the two are almost surely mutually absolutely continuous.

math.PR↗

arXiv2024

Tightness for branching random walk in a space-inhomogeneous random environment

We consider the maximum $M_t$ of branching random walk in a space-inhomogeneous random environment on $\mathbb{Z}$. In this model the branching rate while at some location $x\in\mathbb{Z}$ is randomized in an i.i.d. manner. We prove that there is a centering $\widetilde{m}_t$ depending only on the environment such that $(M_t-\widetilde{m}_t)_{t\ge 0}$ is tight in an annealed sense.

math.PR↗

arXiv2023

Voting models and tightness for a family of recursion equations

We consider recursion equations of the form $u_{n+1}(x)=Q[u_n](x),~n\ge 1,~x\in R$, with a non-local operator $Q[u](x)= g( u\ast q)$, where $g$ is a polynomial, satisfying $g(0)=0$, $g(1)=1$, $g((0,1)) \subseteq (0,1)$, and $q$ is a (compactly supported) probability density with $\ast$ denoting convolution. Motivated by a line of works for nonlinear PDEs initiated by Etheridge, Freeman and Penington (2017), we show that for general $g$, a probabilistic model based on branching random walk can be given to the solution of the recursion, while in case $g$ is also strictly monotone, a probabilistic threshold-based model can be given. In the latter case, we provide a conditional tightness result. We analyze in detail the bistable case and prove for it convergence of the solution shifted around a linear in $n$ centering.

math.PR↗

arXiv2022

Tightness for branching random walk in time-inhomogeneous random environment

We consider a branching random walk in time-inhomogeneous random environment, in which all particles at generation $k$ branch into the same random number of particles $\mathcal{L}_{k+1}\ge 2$, where the $\mathcal{L}_k$, $k\in\mathbb{N}$, are i.i.d., and the increments are standard normal. Let $\mathbb{P}$ denote the law of $(\mathcal{L}_k)_{k\in\mathbb{N}}$, and let $M_n$ denote the position of the maximal particle in generation $n$. We prove that there are $m_n$, which are functions of only $(\mathcal{L}_k)_{k\in\{0,\dots, n\}}$, such that (with regard to $\mathbb{P}$) the sequence $(M_n-m_n)_{n\in\mathbb{N}}$ is tight with high probability.

math.PR↗

arXiv2020

Subsequential tightness for branching random walk in random environment

We consider branching random walk in random environment (BRWRE) and prove the existence of deterministic subsequences along which their maximum, centered at its mean, is tight. This partially answers an open question in arXiv:1711.00852. The method of proof adapts an argument developed by Dekking and Host for branching random walks with bounded increments. The question of tightness without the need for subsequences remains open.

math.PR↗