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Wojciech Cygan

Publications and source records attributed to Wojciech Cygan.

At least 19 recordsLinked to original sources
arXiv2026

Fluctuations for diameter and perimeter of convex hulls of multiple random walks

We study the diameter and perimeter of the convex hull generated by finitely many independent planar random walks whose increments have finite second moments. The large-time fluctuations are governed by the geometry of the polygon formed by the drift vectors. We develop an $L^2$-approximation framework, based on Wald-type maximal central limit theorems, which reduces the asymptotic analysis of the hull to a finite collection of endpoint, maximal-projection, and Brownian support-function terms. For the diameter, we obtain general max-type limit theorems, Gaussian in the case of a unique extremal diametrical pair and typically non-Gaussian when several extremal pairs compete. For the perimeter, we prove a general distributional limit: non-zero extremal drifts contribute maxima of Gaussian projections, while zero-drift extremal walks contribute Brownian support-function terms. The results recover the previously known Gaussian regimes (the case of one or two walks) and identify the non-Gaussian limits in the degenerate and boundary cases left open (even for two walks). We also give $L^2$ approximations of the convex hull by simpler random sets, under Hausdorff and $\ell_1$ metrics on compact convex sets. Our proofs work under the optimal finite second moment assumption.

math.PR↗
arXiv2025

Heat kernels, intrinsic contractivity and ergodicity of discrete-time Markov chains killed by potentials

We study discrete-time Markov chains on countably infinite state spaces, which are perturbed by rather general confining (i.e.\ growing at infinity) potentials. Using a discrete-time analogue of the classical Feynman--Kac formula, we obtain two-sided estimates for the $n$-step heat kernels $u_n(x,y)$ of the perturbed chain. These estimates are of the form $u_n(x,y)\asymp λ_0^nϕ_0(x)\widehatϕ_0(y)+F_n(x,y)$, where $ϕ_0$ (and $\widehatϕ_0$) are the (dual) eigenfunctions for the lowest eigenvalue $λ_0$; the perturbation $F_n(x,y)$ is explicitly given, and it vanishes if either $x$ or $y$ is in a bounded set. The key assumptions are that the chain is uniformly lazy and that the \enquote{direct step property} (DSP) is satisfied. This means that the chain is more likely to move from state $x$ to state $y$ in a single step rather than in two or more steps. Starting from the form of the heat kernel estimate, we define the intrinsic (or ground-state transformed) chains and we introduce time-dependent ultracontractivity notions -- asymptotic and progressive intrinsic ultracontractivity -- which we can link to the growth behaviour of the confining potential; this allows us to consider arbitrarily slow growing potentials. These new notions of ultracontractivity also lead to a characterization of uniform (quasi-)ergodicity of the perturbed and the ground-state transformed Markov chains. At the end of the paper, we give various examples that illustrate how our findings relate to existing models, e.g.\ nearest-neighbour walks on infinite graphs, subordinate processes or non-reversible Markov chains.

math.PR↗
arXiv2024

Asymptotics and geometric flows for a class of nonlocal curvatures

We consider a family of nonlocal curvatures determined through a kernel which is symmetric and bounded from above by a radial and radially non-increasing profile satisfying an integrability condition. It turns out that such definition encompasses various variants of nonlocal curvatures that have already appeared in the literature, including fractional curvature and anisotropic fractional curvature. The main task undertaken in the article is to study the limit behaviour of the introduced nonlocal curvatures under an appropriate limiting procedure. This enables us to recover known asymptotic results e.g. for the fractional curvature and for the anisotropic fractional curvature. For the convergence of anisotropic fractional curvatures we identify the limit object as the nonlocal curvature being the first variation of the related anisotropic fractional perimeter. We also prove existence, uniqueness and stability of viscosity solutions to the corresponding level-set parabolic Cauchy problem formulated in terms of the investigated nonlocal curvature.

math.AP↗
arXiv2024

Alexandrov Theorem for nonlocal curvature

In this article we obtain a nonlocal version of the Alexandrov Theorem which asserts that the only set with sufficiently smooth boundary and of constant nonlocal mean curvature is an Euclidean ball. We consider a general nonlocal mean curvature given by a radial and monotone kernel and we formulate an easy-to-check condition which is necessary and sufficient for the nonlocal version of the Alexandrov Theorem to hold in the treated context. Our definition encompasses numerous examples of various nonlocal mean curvatures that have been already studied in the literature. To prove the main result we obtain a specific formula for the tangential derivative of the nonlocal mean curvature and combine it with an application of the method of moving planes.

math.AP↗
arXiv2024

Bounds on the size of the convex hull of planar Brownian motion and related inverse processes

We establish bounds on expected values of various geometric quantities that describe the size of the convex hull spanned by a path of the standard planar Brownian motion. Expected values of the perimeter and the area of the Brownian convex hull are known explicitly, and satisfactory bounds on the expected value of its diameter can be found in the literature as well. In this work we investigate circumradius and inradius of the Brownian convex hull and obtain lower and upper bounds on their expected values. Our other goal is to find bounds on the related inverse processes (that correspond to the perimeter, area, diameter, circumradius and inradius of the convex hull) which provide us with some information on the speed of growth of the size of the Brownian convex hull.

math.PR↗
arXiv2024

Stable random walks in cones

In this paper we consider a multidimensional random walk killed on leaving a right circular cone with a distribution of increments belonging to the normal domain of attraction of an $α$-stable and rotationally-invariant law with $α\in (0,2)\setminus \{1\}$. Based on Bogdan et al. (2018) describing the tail behaviour of the exit time of $α$-stable process from a cone and using some properties of Martin kernel of the isotropic $α$-stable process, in this paper we construct a positive harmonic function of the discrete time random walk under consideration. Then we find the asymptotic tail of the distribution of the exit time of this random walk from the cone. We also prove the corresponding conditional functional limit theorem.

math.PR↗
arXiv2024

Iterated-logarithm laws for convex hulls of random walks with drift

We establish laws of the iterated logarithm for intrinsic volumes of the convex hull of many-step, multidimensional random walks whose increments have two moments and a non-zero drift. Analogous results in the case of zero drift, where the scaling is different, were obtained by Khoshnevisan. Our starting point is a version of Strassen's functional law of the iterated logarithm for random walks with drift. For the special case of the area of a planar random walk with drift, we compute explicitly the constant in the iterated-logarithm law by solving an isoperimetric problem reminiscent of the classical Dido problem. For general intrinsic volumes and dimensions, our proof exploits a novel zero--one law for functionals of convex hulls of walks with drift, of some independent interest. As another application of our approach, we obtain iterated-logarithm laws for intrinsic volumes of the convex hull of the centre of mass (running average) process associated to the random walk.

math.PR↗
arXiv2022

Asymptotics of non-local perimeters

We introduce a notion of non-local perimeter which is defined through an arbitrary positive Borel measure on $\mathbb{R}^d$ which integrates the function $1\wedge |x|$. Such definition of non-local perimeter encompasses a wide range of perimeters which have been already studied in the literature, including fractional perimeters and anisotropic fractional perimeters. The main part of the article is devoted to the study of the asymptotic behaviour of non-local perimeters. As direct applications we recover well-known convergence results for fractional perimeters and anisotropic fractional perimeters

math.AP↗
arXiv2022

Convex hulls of stable random walks

We consider convex hulls of random walks whose steps belong to the domain of attraction of a stable law in $\mathbb{R}^d$. We prove convergence of the convex hull in the space of all convex and compact subsets of $\mathbb{R}^d$, equipped with the Hausdorff distance, towards the convex hull spanned by a path of the limit stable Lévy process. As an application, we establish convergence of (expected) intrinsic volumes under some mild moment/structure assumptions posed on the random walk.

math.PR↗
arXiv2021

Decay of harmonic functions for discrete time Feynman--Kac operators with confining potentials

We propose and study a certain discrete time counterpart of the classical Feynman--Kac semigroup with a confining potential in countable infinite spaces. For a class of long range Markov chains which satisfy the direct step property we prove sharp estimates for functions which are (sub-, super-)harmonic in infinite sets with respect to the discrete Feynman--Kac operators. These results are compared with respective estimates for the case of a nearest-neighbour random walk which evolves on a graph of finite geometry. We also discuss applications to the decay rates of solutions to equations involving graph Laplacians and to eigenfunctions of the discrete Feynman--Kac operators. We include such examples as non-local discrete Schrödinger operators based on fractional powers of the nearest-neighbour Laplacians and related quasi-relativistic operators. Finally, we analyse various classes of Markov chains which enjoy the direct step property and illustrate the obtained results by examples.

math.PR↗
arXiv2021

CLT for the capacity of the range of stable random walks

In this article, we establish a central limit theorem for the capacity of the range process for a class of $d$-dimensional symmetric $α$-stable random walks with the index satisfying $d > 5α/2$. Our approach is based on controlling the limit behavior of the variance of the capacity of the range process which then allows us to apply the Lindeberg-Feller theorem.

math.PR↗
arXiv2020

Limit theorems for a stable sausage

In this article, we study fluctuations of the volume of a stable sausage defined via a $d$-dimensional rotationally invariant $α$-stable process. As the main results, we establish a functional central limit theorem (in the case when $d/α>3 /2$) with a standard one-dimensional Brownian motion in the limit, and Khintchine's and Chung's laws of the iterated logarithm (in the case when $d/α>9 /5$).

math.PR↗
arXiv2020

Functional CLT for the range of stable random walks

In this note, we establish a functional central limit theorem for the capacity of the range for a class of $α$-stable random walks on the integer lattice $\mathbb{Z}^d$ with $d > 5α/2$. Using similar methods, we also prove an analogous result for the cardinality of the range when $d > 3α/ 2$.

math.PR↗
arXiv2020

Transition probability estimates for subordinate random walks

Let $S_n$ be the simple random walk on the integer lattice $\mathbb{Z}^d$. For a Bernstein function $ϕ$ we consider a random walk $S^ϕ_n$ which is subordinated to $S_n$. Under a certain assumption on the behaviour of $ϕ$ at zero we establish global estimates for the transition probabilities of the random walk $S^ϕ_n$. The main tools that we apply are the parabolic Harnack inequality and appropriate bounds for the transition kernel of the corresponding continuous time random walk.

math.PR↗
arXiv2019

Oscillating heat kernels on ultrametric spaces

Let $(X,d)$ be a proper ultrametric space. Given a measure $m$ on $X$ and a function $B \mapsto C(B)$ defined on the collection of all non-singleton balls $B$ of $X$, we consider the associated hierarchical Laplacian $L=L_{C}\,$. The operator $L$ acts in $\mathcal{L}^{2}(X,m),$ is essentially self-adjoint and has a pure point spectrum. It admits a continuous heat kernel $\mathfrak{p}(t,x,y)$ with respect to $m$. We consider the case when $X$ has a transitive group of isometries under which the operator $L$ is invariant and study the asymptotic behaviour of the function $t\mapsto \mathfrak{p}(t,x,x)=\mathfrak{p}(t)$. It is completely monotone, but does not vary regularly. When $X=\mathbb{Q}_{p}\,$, the ring of $p$-adic numbers, and $L=\mathcal{D}^α $, the operator of \ fractional derivative of order $α,$ we show that $\mathfrak{p}(t)=t^{-1/α}\mathcal{A}% (\log_{p}t)$, where $\mathcal{A}(τ)$ is a continuous non-constant $α$-periodic function. We also study asymptotic behaviour of $\min\mathcal{A}$ and $\max\mathcal{A}$ as the space parameter $p$ tends to $\infty$. When $X=S_{\infty}\,$, the infinite symmetric group, and $L$ is a hierarchical Laplacian with metric structure analogous to $\mathcal{D}^α,$ we show that, contrary to the previous case, the completely monotone function $\mathfrak{p}(t)$ oscillates between two functions $ψ(t)$ and $Ψ(t)$ such that $ψ(t)/Ψ(t)\to 0$ as $t \to \infty\,$.

math.PR↗
arXiv2018

A note on the generalized heat content for Lévy processes

Let $\mathbf{X}=\{X_t\}_{t\geq 0}$ be a Lévy process in $\mathbb{R}^d$ and $Ω$ be an open subset of $\mathbb{R}^d$ with finite Lebesgue measure. The quantity $H (t) = \int_Ω \mathbb{P}^{x} (X_t\in Ω^c) d x$ is called the heat content. In this article we consider its generalized version $H_g^μ(t) = \int_{\mathbb{R}^d}\mathbb{E}^{x} g(X_t)μ( d x )$, where $g$ is a bounded function and $μ$ a finite Borel measure. We study its asymptotic behaviour at zero for various classes of Lévy processes.

math.PR↗
arXiv2018

On recurrence of the multidimensional Lindley process

A Lindley process arises from classical studies in queueing theory and it usually reflects waiting times of customers in single server models. In this note we study recurrence of its higher dimensional counterpart under some mild assumptions on the tail behaviour of the underlying random walk. There are several links between the Lindley process and the associated random walk and we build upon such relations. We apply a method related to discrete subordination for random walks on the integer lattice together with various facts from the theory of fluctuations of random walks.

math.PR↗