arXiv subjects

Franziska Kühn

Publications and source records attributed to Franziska Kühn.

At least 19 recordsLinked to original sources

arXiv2025

Enhancement of magnon flux toward a Bose-Einstein condensate

We present a combined theoretical and experimental study of angle-dependent parametric pumping of magnons in Yttrium Iron Garnet films, with a focus on the mechanisms that transfer parametrically injected magnons toward the spectral minimum where Bose-Einstein condensation occurs. Using a classical Hamiltonian formalism, we analyze the threshold conditions for parametric instability as a function of the angle between the microwave pumping field and the external magnetic field, continuously tracing the transition between parallel and transverse pumping. We also describe two competing four-magnon scattering mechanisms that transfer parametric magnons toward the bottom of their frequency spectrum: The step-by-step Kolmogorov-Zakharov cascade, which is allowed for all magnetic field values, and the kinetic instability mechanisms that provide a much more efficient single-step channel in transferring magnons directly to the lowest-energy states, but occurs within specific regions of the pumping angle and the external magnetic field where the conservation laws permit it. In the experimental part, we employ microfocused Brillouin light scattering spectroscopy in combination with a vector magnet, allowing for angle-resolved mapping of the magnon population spectrum under controlled pumping angle. We observe that transverse pumping, although characterized by a higher instability threshold, yields a markedly stronger population at the spectral minimum compared to parallel pumping. These observations demonstrate that the kinetic instability channel plays a dominant role in transferring magnons to the spectral minimum under such conditions. These results reveal the crucial role of pumping geometry in shaping the magnon distribution and provide guidelines for optimizing the flux of magnons into the condensate, thereby advancing the control of magnon Bose-Einstein condensation in magnetic insulators.

cond-mat.quant-gas↗

arXiv2023

Local temperature control of magnon frequency and direction of supercurrents in a magnon Bose-Einstein condensate

The creation of temperature variations in magnetization, and hence in the frequencies of the magnon spectrum in laser-heated regions of magnetic films, is an important method for studying Bose-Einstein condensation of magnons, magnon supercurrents, Bogoliubov waves, and similar phenomena. In our study, we demonstrate analytically, numerically, and experimentally that, in addition to the magnetization variations, it is necessary to consider the connected variations of the demagnetizing field. In case of a heat induced local minimum of the saturation magnetization, the combination of these two effects results in a local increase in the minimum frequency value of the magnon dispersion at which the Bose-Einstein condensate emerges. As a result, a magnon supercurrent directed away from the hot region is formed.

cond-mat.quant-gas↗

arXiv2023

Rapid-prototyping of microscopic thermal landscapes in Brillouin light scattering spectroscopy

Since temperature and its spatial and temporal variations affect a wide range of physical properties of material systems, they can be used to create reconfigurable spatial structures of various types in physical and biological objects. This paper presents an experimental optical setup for creating tunable two-dimensional temperature patterns on a micrometer scale. As an example of its practical application, we have produced temperature-induced magnetization landscapes in ferrimagnetic yttrium iron garnet films and investigated them using micro-focused Brillouin light scattering spectroscopy. It is shown that, due to the temperature dependence of the magnon spectrum, temperature changes can be visualized even for microscale thermal patterns.

cond-mat.quant-gas↗

arXiv2023

Maximal Inequalities and Some Applications

A maximal inequality is an inequality which involves the (absolute) supremum $\sup_{s\leq t}|X_{s}|$ or the running maximum $\sup_{s\leq t}X_{s}$ of a stochastic process $(X_t)_{t\geq 0}$. We discuss maximal inequalities for several classes of stochastic processes with values in an Euclidean space: Martingales, Lévy processes, Lévy-type - including Feller processes, (compound) pseudo Poisson processes, stable-like processes and solutions to SDEs driven by a Lévy process -, strong Markov processes and Gaussian processes. Using the Burkholder-Davis-Gundy inequalities we als discuss some relations between maximal estimates in probability and the Hardy-Littlewood maximal functions from analysis. This paper has been accepted for publication in Probability Surveys

math.PR↗

arXiv2022

Lévy Processes, Generalized Moments and Uniform Integrability

We give new proofs of certain equivalent conditions for the existence of generalized moments of a Lévy process $(X_t)_{t\geq 0}$; in particular, the existence of a generalized $g$-moment is equivalent to the uniform integrability of $(g(X_t))_{t\in [0,1]}$. As a consequence, certain functions of a Lévy process which are integrable and local martingales are already true martingales. Our methods extend to moments of stochastically continuous additive processes, and we give new, short proofs for the characterization of lattice distributions and the transience of Lévy processes.

math.PR↗

arXiv2021

For which functions are $f(X_t)-\mathbb{E} f(X_t)$ and $g(X_t)/\mathbb{E} g(X_t)$ martingales?

Let $X=(X_t)_{t\geq 0}$ be a one-dimensional Lévy process such that each $X_t$ has a $C^1_b$-density w.r.t. Lebesgue measure and certain polynomial or exponential moments. We characterize all polynomially bounded functions $f:\mathbb{R}\to\mathbb{R}$, and exponentially bounded functions $g:\mathbb{R}\to (0,\infty)$, such that $f(X_t)-\mathbb{E} f(X_t)$, resp. $g(X_t)/\mathbb{E} g(X_t)$, are martingales.

math.PR↗

arXiv2021

Upper functions for sample paths of Lévy(-type) processes

We study the small-time asymptotics of sample paths of Lévy processes and Lévy-type processes. Namely, we investigate under which conditions the limit $$\limsup_{t \to 0} \frac{1}{f(t)} |X_t-X_0|$$ is finite resp.\ infinite with probability $1$. We establish integral criteria in terms of the infinitesimal characteristics and the symbol of the process. Our results apply to a wide class of processes, including solutions to Lévy-driven SDEs and stable-like processes. For the particular case of Lévy processes, we recover and extend earlier results from the literature. Moreover, we present a new maximal inequality for Lévy-type processes, which is of independent interest.

math.PR↗

arXiv2021

Feller Generators with measurable lower order terms

We study perturbations of Feller generators under `lower order terms' with measurable coefficients. We investigate which properties of the original semigroup -- such as positivity, conservativeness and the Feller property -- are passed to the perturbed semigroup. We give several examples and discuss applications in the theory of martingale problems and stochastic differential equations with measurable coefficients.

math.PR↗

arXiv2021

Convolution inequalities for Besov and Triebel--Lizorkin spaces, and applications to convolution semigroups

We establish convolution inequalities for Besov spaces $B_{p,q}^s$ and Triebel--Lizorkin spaces $F_{p,q}^s$. As an application, we study the mapping properties of convolution semigroups, considered as operators on the function spaces $A_{p,q}^s$, $A \in \{B,F\}$. Our results apply to a wide class of convolution semigroups including the Gauß--Weierstraß semigroup, stable semigroups and heat kernels for higher-order powers of the Laplacian $(-Δ)^m$, and we can derive various caloric smoothing estimates.

math.FA↗

arXiv2020

A Liouville theorem for Lévy generators

Under mild assumptions, we establish a Liouville theorem for the "Laplace" equation $Au=0$ associated with the infinitesimal generator $A$ of a Lévy process: If $u$ is a weak solution to $Au=0$ which is at most of (suitable) polynomial growth, then $u$ is a polynomial. As a by-product, we obtain new regularity estimates for semigroups associated with Lévy processes.

math.PR↗

arXiv2020

Strong convergence of the Euler--Maruyama approximation for a class of Lévy-driven SDEs

Consider the following stochastic differential equation (SDE) $$dX_t = b(t,X_{t-}) \, dt+ dL_t, \quad X_0 = x,$$ driven by a $d$-dimensional Lévy process $(L_t)_{t \geq 0}$. We establish conditions on the Lévy process and the drift coefficient $b$ such that the Euler--Maruyama approximation converges strongly to a solution of the SDE with an explicitly given rate. The convergence rate depends on the regularity of $b$ and the behaviour of the Lévy measure at the origin. As a by-product of the proof, we obtain that the SDE has a pathwise unique solution. Our result covers many important examples of Lévy processes, e.g. isotropic stable, relativistic stable, tempered stable and layered stable.

math.PR↗

arXiv2020

Interior Schauder estimates for elliptic equations associated with Lévy operators

We study the local regularity of solutions $f$ to the integro-differential equation $$ Af=g \quad \text{in $U$}$$ associated with the infinitesimal generator $A$ of a Lévy process $(X_t)_{t \geq 0}$. Under the assumption that the transition density of $(X_t)_{t \geq 0}$ satisfies a certain gradient estimate, we establish interior Schauder estimates for both pointwise and weak solutions $f$. Our results apply for a wide class of Lévy generators, including generators of stable Lévy processes and subordinated Brownian motions.

math.PR↗

arXiv2019

On infinitesimal generators of sublinear Markov semigroups

We establish a Dynkin formula and a Courrège-von Waldenfels theorem for sublinear Markov semigroups. In particular, we show that any sublinear operator $A$ on $C_c^{\infty}(\mathbb{R}^d)$ satisfying the positive maximum principle can be represented as supremum of a family of pseudo-differential operators: $$Af(x) = \sup_{α\in I} (-q_α(x,D) f)(x).$$ As an immediate consequence, we obtain a representation formula for infinitesimal generators of sublinear Markov semigroups with a sufficiently rich domain. We give applications in the theory of non-linear Hamilton--Jacobi--Bellman equations and Lévy processes for sublinear expectations.

math.PR↗

arXiv2019

Schauder estimates for Poisson equations associated with non-local Feller generators

We show how Hölder estimates for Feller semigroups can be used to obtain regularity results for solutions to the Poisson equation $Af=g$ associated with the (extended) infinitesimal generator $A$ of a Feller process. The regularity of $f$ is described in terms of Hölder-Zygmund spaces of variable order and, moreover, we establish Schauder estimates. Since Hölder estimates for Feller semigroups have been intensively studied in the last years, our results apply to a wide class of Feller processes, e.g. random time changes of Lévy processes and solutions to Lévy-driven stochastic differential equations. Most prominently, we establish Schauder estimates for the Poisson equation associated with the fractional Laplacian of variable order. As a by-product, we obtain new regularity estimates for semigroups associated with stable-like processes.

math.PR↗

arXiv2019

Viscosity solutions to Hamilton-Jacobi-Bellman equations associated with sublinear Lévy(-type) processes

Using probabilistic methods we study the existence of viscosity solutions to non-linear integro-differential equations $$\partial_t u(t,x) - \sup_{α\in I} \bigg( b_α(x) \cdot \nabla_x u(t,x) + \frac{1}{2} \text{tr}\left(Q_α(x) \cdot \nabla^2_x u(t,x)\right) +\int_{y \neq 0} \big(u(t,x+y)-u(t,x)-\nabla_x u(t,x) \cdot h(y) \big) \, ν_α(x,dy) \bigg) = 0$$ with initial condition $u(0,x)= φ(x)$; here $(b_α(x),Q_α(x),ν_α(x,dy))$, $α\in I$, $x \in \mathbb{R}^d$, is a family of Lévy triplets and $h$ is some truncation function. The solutions, which we construct, are of the form $u(t,x) = T_t φ(x)$ for a sublinear Markov semigroup $(T_t)_{t \geq 0}$ with representation $$T_t φ(x) = \mathcal{E}^x φ(X_t):= \sup_{\mathbb{P} \in \mathfrak{P}_x} \int_Ω φ(X_t) \, d\mathbb{P}$$ where $(X_t)_{t \geq 0}$ is a stochastic process and $\mathfrak{P}_x$, $x \in \mathbb{R}^d$, are families of probability measures. The key idea is to exploit the connection between sublinear Markov semigroups and the associated Kolmogorov backward equation. In particular, we obtain new existence and uniqueness results for viscosity solutions to Kolmogorov backward equations associated with Lévy(-type) processes for sublinear expectations and Feller processes on classical probability spaces.

math.PR↗

arXiv2019

Schauder estimates for equations associated with Lévy generators

We study the regularity of solutions to the integro-differential equation $Af-λf=g$ associated with the infinitesimal generator $A$ of a Lévy process. We show that gradient estimates for the transition density can be used to derive Schauder estimates for $f$. Our main result allows us to establish Schauder estimates for a wide class of Lévy generators, including generators of stable Lévy processes and subordinate Brownian motions. Moreover, we obtain new insights on the (domain of the) infinitesimal generator of a Lévy process whose characteristic exponent $ψ$ satisfies $\text{Re} \, ψ(ξ) \asymp |ξ|^α$ for large $|ξ|$. We discuss the optimality of our results by studying in detail the domain of the infinitesimal generator of the Cauchy process.

math.PR↗

arXiv2018

On the domain of fractional Laplacians and related generators of Feller processes

In this paper we study the domain of stable processes, stable-like processes and more general pseudo- and integro-differential operators which naturally arise both in analysis and as infinitesimal generators of Lévy- and Lévy-type (Feller) processes. In particular we obtain conditions on the symbol of the operator ensuring that certain (variable order) Hölder and Hölder-Zygmund spaces are in the domain. We use tools from probability theory to investigate the small-time asymptotics of the generalized moments of a Lévy or Lévy-type process $(X_t)_{t \geq 0}$, \begin{equation*} \lim_{t \to 0} \frac 1t\left(\mathbb{E}^x f(X_t)-f(x)\right), \quad x\in\mathbb{R}^d, \end{equation*} for functions $f$ which are not necessarily bounded or differentiable. The pointwise limit exists for fixed $x \in \mathbb{R}^d$ if $f$ satisfies a Hölder condition at $x$. Moreover, we give sufficient conditions which ensure that the limit exists uniformly in the space of continuous functions vanishing at infinity. As an application we prove that the domain of the generator of $(X_t)_{t \geq 0}$ contains certain Hölder spaces of variable order. Our results apply, in particular, to stable-like processes, relativistic stable-like processes, solutions of Lévy-driven SDEs and Lévy processes.

math.PR↗

arXiv2018

A probabilistic proof of Schoenberg's theorem

Assume that $g(|ξ|^2)$, $ξ\in\mathbb{R}^k$, is for every dimension $k\in\mathbb{N}$ the characteristic function of an infinitely divisible random variable $X^k$. By a classical result of Schoenberg $f:=-\log g$ is a Bernstein function. We give a simple probabilistic proof of this result starting from the observation that $X^k = X_1^k$ can be embedded into a Lévy process $(X_t^k)_{t\geq 0}$ and that Schoenberg's theorem says that $(X_t^k)_{t\geq 0}$ is subordinate to a Brownian motion. A key ingredient of our proof are concrete formulae which connect the transition densities, resp., Lévy measures of subordinated Brownian motions across different dimensions. As a by-product of our proof we obtain a gradient estimate for the transition semigroup of a subordinated Brownian motion.

math.PR↗