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Michael A. Högele

Publications and source records attributed to Michael A. Högele.

At least 19 recordsLinked to original sources
arXiv2026

Deviation frequencies of Brownian path property approximations

This case study proposes a.s.~convergence quantifications of many classical sample path property approximations of Brownian motion in terms of the tradeoff between a.s.~rates and the integrability of the modulus of convergence, as well as the deviation frequencies. This includes Lévy's construction of Brownian motion, the Kolmogorov-Chentsov (and the Kolmogorov-Totoki) continuity theorem, Lévy's modulus of continuity, the Paley-Wiener-Zygmund theorem, the a.s.~approximation of the quadratic variation as well as the laws of the iterated logarithm by Khinchin, Chung and Strassen, among others.

math.PR↗
arXiv2026

Ergodicity bounds in the Sliced Wasserstein distance for Schur stable autoregressive processes

Explicit calculations in dimension one show for Schur stable autoregressive processes with standard Gaussian noise that the ergodic convergence in the Wasserstein-$2$ distance is essentially given by the sum of the mean, which decays exponentially, and the standard deviation, which decays with twice the speed. This paper starts by showing new upper and lower multivariate affine transport bounds for the Wasserstein-$r$ distance for $r$ greater and equal to $1$. These bounds allow to formulate a novel sufficient (non-Gaussian) ergodic interpolation condition for the mentioned mean-variance behavior to take place in case of more general Schur stable multivariate autoregressive processes. All ergodic estimates are non-asymptotic with completely explicit constants. The main applications are precise thermalization bounds for Schur stable $\mathsf{AR}(p)$ and $\mathsf{ARMA}(p,q)$ models in Wasserstein and Sliced Wasserstein distance. In the sequel we establish with the help of coupling techniques explicit upper and lower exponential bounds for more general multivariate Schur stable autoregressive processes. This includes parallel sampling and the convergence of the empiricial means. The utility of our results in particular for the Sliced Wasserstein distance are confirmed by multivariate numerical experiments.

math.PR↗
arXiv2025

A Schauder-Tychonoff fixed-point approach for nonlinear Lévy driven reaction-diffusion systems

We show a stochastic version of the Schauder-Tychonoff fixed point theorem which yields a solution of the martingale problem for a class of systems of nonlinear reaction-diffusion equations driven by a cylindrical Wiener process and a Poisson random measure with certain moments. By this type of theorem one can solve systems by linearization which have a possibly unbounded, non-dissipative and non-coercive nonlinearity.

math.PR↗
arXiv2025

Weak solutions for coupled reaction-diffusion systems with pattern formation by a stochastic fixed point theorem

Chemical and biochemical reactions can exhibit surprisingly different behaviours, ranging from multiple steady-state solutions to oscillatory solutions and chaotic behaviours. These types of systems are often modelled by a system of reaction-diffusion equations coupled by a nonlinearity. In the article, we study the existence of stochastically perturbed equations of this type. In particular, we show the existence of a probabilitic weak solution of the following stochastic system \begin{align*} \dot {u} & = r_1\,Δu+ a_1\, u + b_1 -c_1\, u\cdot v^q+σ_1\, g_1(u)\circ \dot W_1, \\ \dot{v} & = r_2 \,A v + a_2\, v + b_2 +c_2\, u\cdot v^q + σ_2\, g_2(v)\circ \dot W_2, \end{align*} where $r_i,b_i,c_i, σ_i>0$, $a_i\in\mathbb{R}$, and $g_i$ are linear, $i=1,2$, and the exponent $q\geq 1$. The operator $A=-(-Δ)^{\aleph/2}$ is a fractional power of the Laplacian, $1<\aleph \le2$. The main result is obtained by a Schauder-Tychonoff type fixed point theorem for the controlled versions of the laws of the respective (infinite dimensional) Ornstein-Uhlenbeck system, from which we infer the existence of a weak solution of the coupled system.

math.AP↗
arXiv2025

Large deviations for light-tailed Lévy bridges on short time scales

Let $L = (L(t))_{t\geq 0}$ be a multivariate Lévy process with Lévy measure $ν(dy) = \exp(-f(|y|)) dy$ for a smoothly regularly varying function $f$ of index $α>1$. The process $L$ is renormalized as $X^\varepsilon(t) = \varepsilon L(r_\varepsilon t)$, $t\in [0, T]$, for a scaling parameter $r_\varepsilon= o(\varepsilon^{-1})$, as $\varepsilon \to 0$. We study the behavior of the bridge $Y^{\varepsilon, x}$ of the renormalized process $X^\varepsilon$ conditioned on the event $X^\varepsilon(T) = x$ for a given end point $x\neq 0$ and end time $T>0$ in the regime of small $\varepsilon$. Our main result is a sample path large deviations principle (LDP) for $Y^{\varepsilon, x}$ with a specific speed function $S(\varepsilon)$ and an entropy-type rate function $I_{x}$ on the Skorokhod space in the limit $\varepsilon \rightarrow 0+$. We show that the asymptotic energy minimizing path of $Y^{\varepsilon, x}$ is the linear parametrization of the straight line between $0$ and $x$, while all paths leaving this set are exponentially negligible. We also infer a LDP for the asymptotic number of jumps and establish asymptotic normality of the jump increments of $Y^{\varepsilon, x}$. Since on these short time scales $r_\varepsilon = o(\varepsilon^{-1})$) direct LDP methods cannot be adapted we use an alternative direct approach based on convolution density estimates of the marginals $X^{\varepsilon}(t)$, $t\in [0, T]$,for which we solve a specific nonlinear functional equation.

math.PR↗
arXiv2025

On the tradeoff between almost sure error tolerance and mean deviation frequency in martingale convergence

In this article we quantify almost sure martingale convergence theorems in terms of the tradeoff between asymptotic almost sure rates of convergence (error tolerance) and the respective modulus of convergence. For this purpose we generalize {an} elementary quantitative version of the first Borel-Cantelli lemma on the statistics of the deviation frequencies (error incidence), which was recently established by the authors. First we study martingale convergence in $L^2$, and in the setting of the Azuma-Hoeffding inequality. In a second step we study the strong law of large numbers for martingale differences in two settings: uniformly bounded increments in $L^p$, $p\geq 2$, using the respective Baum-Katz-Stoica theorems, and uniformly bounded exponential moments with the help of the martingale estimates by Lesigne and Volný. We also present applications for the tradeoff for the multicolor generalized Pólya urn process, the Generalized Chinese restaurant process, statistical M-estimators, as well as the a.s.~excursion frequencies of the Galton-Watson branching process. Finally, we relate the tradeoff concept to the convergence in the Ky Fan metric.

math.PR↗
arXiv2025

Cut-off phenomenon and asymptotic mixing for multivariate general linear processes

The small noise cut-off phenomenon in continuous time and space has been studied in the recent literature for the linear and non-linear stable Langevin dynamics with additive Lévy drivers - understood as abrupt thermalization of the system along a particular time scale to its dynamical equilibrium - both for the total variation distance and the Wasserstein distance. The main result of this article establishes sufficient conditions for the window and profile cut-off phenomenon, which are flexible enough to cover the renormalized (non-Markovian) Ornstein--Uhlenbeck process driven by fractional Brownian motion and a large class of Gaussian and non-Gaussian, homogeneous and non-homogeneous drivers with (possible) finite second moments. The sufficient conditions are stated both for the total variation distance and the Wasserstein distance. Important examples are the multidimensional fractional Ornstein--Uhlenbeck process, the empirical sampling process of a fractional Ornstein--Uhlenbeck process, an Ornstein--Uhlenbeck processes driven by an Ornstein--Uhlenbeck process and the inhomogeneous Ornstein--Uhlenbeck process arising in simulated annealing.

math.PR↗
arXiv2024

Cutoff ergodicity bounds in Wasserstein distance for a viscous energy shell model with Lévy noise

This article establishes explicit non-asymptotic ergodic bounds in the renormalized Wasserstein-Kantorovich-Rubinstein (WKR) distance for a viscous energy shell lattice model of turbulence with random energy injection. The system under consideration is driven either by a Brownian motion, a symmetric $α$-stable Lévy process, a stationary Gaussian or $α$-stable Ornstein-Uhlenbeck process, or by a general Lévy process with second moments. The obtained non-asymptotic bounds establish asymptotically abrupt thermalization. The analysis is based on the explicit representation of the solution of the system in terms of convolutions of Bessel functions.

math-ph↗
arXiv2023

Ergodicity bounds for stable Ornstein-Uhlenbeck systems in Wasserstein distance with applications to cutoff stability

This article establishes cutoff stability also known as abrupt thermalization for generic multidimensional Hurwitz stable Ornstein-Uhlenbeck systems with (possibly degenerate) Lévy noise at fixed noise intensity. The results are based on several ergodicity quantitative lower and upper bounds some of which make use of the recently established shift linearity property of the Wasserstein-Kantorovich-Rubinstein distance by the authors. It covers such irregular systems like Jacobi chains and more general networks of coupled harmonic oscillators with a heat bath (including Lévy excitations) at constant temperature on the outer edges and the so-called Brownian gyrator.

math.PR↗
arXiv2022

Stochastic n-point D-bifurcations of stochastic Lévy flows and their complexity on finite spaces

This article refines the classical notion of a stochastic D-bifurcation to the respective family of n-point motions for homogeneous Markovian stochastic semiflows, such as stochastic Brownian flows of homeomorphisms, and their generalizations. This notion essentially detects at which level $k\leq n$ the support of the invariant measure of the k-point bifurcation has more than one connected component. Stochastic Brownian flows and their invariant measures which were shown by Kunita (1990) to be rigid, in the sense of being uniquely determined by the $1$-and $2$-point motions, and hence only stochastic n-point bifurcation of level $n=1$ or $n=2$ can occur. For general homogeneous stochastic Markov semiflows this turns out to be false. This article constructs minimal examples of where this rigidity is false in general on finite space and studies the complexity of the resulting n-point bifurcations.

math.PR↗
arXiv2022

The cutoff phenomenon in Wasserstein distance for nonlinear stable Langevin systems with small Lévy noise

This article establishes the cutoff phenomenon in the Wasserstein distance for systems of nonlinear ordinary differential equations with a unique coercive stable fixed point subject to general additive Markovian noise in the limit of small noise intensity. This result generalizes the results shown in Barrera, Högele, Pardo (EJP2021) in a more restrictive setting of Blumenthal-Getoor index $α>3/2$ to the formulation in Wasserstein distance, which allows to cover the case of general Lévy processes with some given moment. The main proof techniques are based on the close control of the errors in a version of the Hartman-Grobman theorem and the adaptation of the linear theory established in Barrera, Högele, Pardo (JSP2021). In particular, they rely on the precise asymptotics of the nonlinear flow and the nonstandard shift linearity property of the Wasserstein distance, which is established by the authors in (JSP2021). Main examples are the Fermi-Pasta-Ulam-Tsingou gradient flow and coercive nonlinear oscillators subject to small (and possibly degenerate) Brownian or arbitrary $α$-stable noise.

math.PR↗
arXiv2021

The cutoff phenomenon in total variation for nonlinear Langevin systems with small layered stable noise

This paper provides an extended case study of the cutoff phenomenon for a prototypical class of nonlinear Langevin systems with a single stable state perturbed by an additive pure jump Lévy noise of small amplitude $\varepsilon>0$, where the driving noise process is of layered stable type. Under a drift coercivity condition the associated family of processes $X^\varepsilon$ turns out to be exponentially ergodic with equilibrium distribution $μ^{\varepsilon}$ in total variation distance which extends a result from Peng and Zhang (2018) to arbitrary polynomial moments. The main results establish the cutoff phenomenon with respect to the total variation, under a sufficient smoothing condition of Blumenthal-Getoor index $α>3/2$. That is to say, in this setting we identify a deterministic time scale $\mathfrak{t}_{\varepsilon}^{\mathrm{cut}}$ satisfying $\mathfrak{t}_ \varepsilon^{\mathrm{cut}} \rightarrow \infty$, as $\varepsilon \rightarrow 0$, and a respective time window, $\mathfrak{t}_\varepsilon^{\mathrm{cut}} \pm o(\mathfrak{t}_\varepsilon^{\mathrm{cut}})$, during which the total variation distance between the current state and its equilibrium $μ^{\varepsilon}$ essentially collapses as $\varepsilon$ tends to zero. In addition, we extend the dynamical characterization under which the latter phenomenon can be described by the convergence of such distance to a unique profile function first established in Barrera and Jara (2020) to the Lévy case for nonlinear drift. This leads to sufficient conditions, which can be verified in examples, such as gradient systems subject to small symmetric $α$-stable noise for $α>3/2$. The proof techniques differ completely from the Gaussian case due to the absence of respective Girsanov transforms which couple the nonlinear equation and the linear approximation asymptotically even for short times.

math.PR↗
arXiv2021

Cutoff thermalization for Ornstein-Uhlenbeck systems with small Lévy noise in the Wasserstein distance

This article establishes cutoff thermalization (also known as the cutoff phenomenon) for a class of generalized Ornstein-Uhlenbeck systems $(X^\varepsilon_t(x))_{t\geqslant 0}$ with $\varepsilon$-small additive Lévy noise and initial value $x$. The driving noise processes include Brownian motion, $α$-stable Lévy flights, finite intensity compound Poisson processes, and red noises, and may be highly degenerate. Window cutoff thermalization is shown under mild generic assumptions; that is, we see an asymptotically sharp $\infty/0$-collapse of the renormalized Wasserstein distance from the current state to the equilibrium measure $μ^\varepsilon$ along a time window centered on a precise $\varepsilon$- and $x$-dependent time scale $t_\varepsilon^x$. In many interesting situations such as reversible (Lévy) diffusions it is possible to prove the existence of an explicit, universal, deterministic cutoff thermalization profile. That is, for generic initial data $x$ we obtain the stronger result $\mathcal{W}_p(X^\varepsilon_{t_\varepsilon + r}(x), μ^\varepsilon) \cdot \varepsilon^{-1} \rightarrow K\cdot e^{-q r}$ as $\varepsilon \rightarrow 0$ for any $r\in \mathbb{R}$, some spectral constants $K, q>0$ and any $p\geqslant 1$ whenever the distance is finite. The existence of this limit is characterized by the absence of non-normal growth patterns in terms of an orthogonality condition on a computable family of generalized eigenvectors of $\mathcal{Q}$. Precise error bounds are given. Using these results, this article provides a complete discussion of the cutoff phenomenon for the classical linear oscillator with friction subject to $\varepsilon$-small Brownian motion or $α$-stable Lévy flights. Furthermore, we cover the highly degenerate case of a linear chain of oscillators in a generalized heat bath at low temperature.

math.PR↗
arXiv2020

The Kramers problem for SDEs driven by small, accelerated Lévy noise with exponentially light jumps

We establish Freidlin-Wentzell results for a nonlinear ordinary differential equation starting close to the stable state $0$, say, subject to a perturbation by a stochastic integral which is driven by an $\varepsilon$-small and $(1/\varepsilon)$-accelerated Lévy process with exponentially light jumps. For this purpose we derive a large deviations principle for the stochastically perturbed system using the weak convergence approach developed by Budhiraja, Dupuis, Maroulas and collaborators in recent years. In the sequel we solve the associated asymptotic first escape problem from the bounded neighborhood of $0$ in the limit as $\varepsilon \rightarrow 0$ which is also known as the Kramers problem in the literature.

math.PR↗
arXiv2019

The first exit problem of reaction-diffusion equations for small multiplicative Lévy noise

This article studies the dynamics of a nonlinear dissipative reaction-diffusion equation with well-separated stable states which is perturbed by infinite-dimensional multiplicative Lévy noise with a regularly varying component at intensity $ε>0$. The main results establish the precise asymptotics of the first exit times and locus of the solution $X^ε$ from the domain of attraction of a deterministic stable state, in the limit as $ε\rightarrow 0$. In contrast to the exponential growth for respective Gaussian perturbations the exit times grow essentially as a power function of the noise intensity as $ε\rightarrow 0$ with the exponent given as the tail index $-α$, $α>0,$ of the Lévy measure, analogously to the case of additive noise in Debussche et al (2013). In this article we substantially improve their quadratic estimate of the small jump dynamics and derive a new exponential estimate of the stochastic convolution for stochastic Lévy integrals with bounded jumps based on the recent pathwise Burkholder-Davis-Gundy inequality by Siorpaes (2018). This allows to cover perturbations with general tail index $α>0$, multiplicative noise and perturbations of the linear heat equation. In addition, our convergence results are probabilistically strongest possible. Finally, we infer the metastable convergence of the process on the common time scale $t/ε^α$ to a Markov chain switching between the stable states of the deterministic dynamical system.

math.PR↗
arXiv2019

The first passage problem for stable linear delay equations perturbed by power law Lévy noise

This article studies a linear scalar delay differential equation subject to small multiplicative power tail Lévy noise. We solve the first passage (the Kramers) problem with probabilistic methods and discover an asymptotic loss of memory in this non-Markovian system. Furthermore, the mean exit time increases as the power of the small noise amplitude, whereas the pre-factor accounts for memory effects. In particular, we discover a non-linear delay-induced exit acceleration due to a non-normal growth phenomenon. Our results are illustrated in the well-known linear delay oscillator driven by $α$-stable Lévy flights.

math.PR↗
arXiv2018

A strong averaging principle for Lévy diffusions in foliated spaces with unbounded leaves

This article extends a strong averaging principle for Lévy diffusions which live on the leaves of a foliated manifold subject to small transversal Lévy type perturbation to the case of non-compact leaves. The main result states that the existence of $p$-th moments of the foliated Lévy diffusion for $p\geq 2$ and an ergodic convergence of its coefficients in $L^p$ implies the strong $L^p$ convergence of the fast perturbed motion on the time scale $t/ε$ to the system driven by the averaged coefficients. In order to compensate the non-compactness of the leaves we use an estimate of the dynamical system for each of the increments of the canonical Marcus equation derived in da Costa and Hoegele (2017), the boundedness of the coefficients in $L^p$ and a nonlinear Gronwall-Bihari type estimate. The price for the non-compactness are slower rates of convergence, given as $p$-dependent powers of $ε$ strictly smaller than $1/4$.

math.DS↗