arXiv subjects

Martin Friesen

Publications and source records attributed to Martin Friesen.

At least 37 records · Page 2Linked to original sources

arXiv2020

On uniqueness and stability for the Enskog equation

The time-evolution of a moderately dense gas in a vacuum is described in classical mechanics by a particle density function obtained from the Enskog equation. Based on a McKean-Vlasov stochastic equation with jumps, the associated stochastic process was recently studied in \cite{ARS17}. The latter work was extended in \cite{FRS18} to the case of general hard and soft potentials without Grad's angular cut-off assumption. By the introduction of a shifted distance that exactly compensates for the free transport term that accrues in the spatially inhomogeneous setting, we prove in this work an inequality on the Wasserstein distance for any two measure-valued solutions to the Enskog equation. As a particular consequence, we find sufficient conditions for the uniqueness and continuous-dependence on initial data for solutions to the Enskog equation applicable to hard and soft potentials without angular cut-off.

math.AP↗

arXiv2019

On the anisotropic stable JCIR process

We investigate the anisotropic stable JCIR process which is a multi-dimensional extension of the stable JCIR process but also a multi-dimensional analogue of the classical JCIR process. We prove that the heat kernel of the anisotropic stable JCIR process exists and it satisfies an a-priori bound in a weighted anisotropic Besov norm. Based on this regularity result we deduce the strong Feller property and prove, for the subcritical case, exponential ergodicity in total variation. Also, we show that in the one-dimensional case the corresponding heat kernel is smooth.

math.PR↗

arXiv2019

Ergodicity of affine processes on the cone of symmetric positive semidefinite matrices

This article investigates the long-time behavior of conservative affine processes on the cone of symmetric positive semidefinite $d\times d$-matrices. In particular, for conservative and subcritical affine processes on this cone we show that a finite $\log$-moment of the state-independent jump measure is sufficient for the existence of a unique limit distribution. Moreover, we study the convergence rate of the underlying transition kernel to the limit distribution: firstly, in a specific metric induced by the Laplace transform and secondly, in the Wasserstein distance under a first moment assumption imposed on the state-independent jump measure and an additional condition on the diffusion parameter.

math.PR↗

arXiv2019

Nonlinear perturbations of evolution systems in scales of Banach spaces

A variant of the abstract Cauchy-Kovalevskaya theorem is considered. We prove existence and uniqueness of classical solutions to the nonlinear, non-autonomous initial value problem \[ \frac{du(t)}{dt} = A(t)u(t) + B(u(t),t), \ \ u(0) = x \] in a scale of Banach spaces. Here $A(t)$ is the generator of an evolution system acting in a scale of Banach spaces and $B(u,t)$ obeys an Ovcyannikov-type bound. Continuous dependence of the solution with respect to $A(t)$, $B(u,t)$ and $x$ is proved. The results are applied to the Kimura-Maruyama equation for the mutation-selection balance model. This yields a new insight in the construction and uniqueness question for nonlinear Fokker-Planck equations related with interacting particle systems in the continuum.

math.FA↗

arXiv2019

Stochastic equation and exponential ergodicity in Wasserstein distances for affine processes

This work is devoted to the study of conservative affine processes on the canonical state space $D = $R_+^m \times \R^n$, where $m + n > 0$. We show that each affine process can be obtained as the pathwise unique strong solution to a stochastic equation driven by Brownian motions and Poisson random measures. Then we study the long-time behavior of affine processes, i.e., we show that under first moment condition on the state-dependent and log-moment conditions on the state-independent jump measures, respectively, each subcritical affine process is exponentially ergodic in a suitably chosen Wasserstein distance. Moments of affine processes are studied as well.

math.PR↗

arXiv2019

Long-time behavior for subcritical measure-valued branching processes with immigration

In this work we study the long-time behavior for subcritical measure-valued branching processes with immigration on the space of tempered measures. Under some reasonable assumptions on the spatial motion, the branching and immigration mechanisms, we prove the existence and uniqueness of an invariant measure for the corresponding Markov transition semigroup. Moreover, we show that it converges with exponential rate to the unique invariant measure in the Wasserstein distance as well as in a distance defined in terms of Laplace transforms. Finally, we consider an application of our results to super-Lévy processes as well as branching particle systems on the lattice with noncompact spins.

math.PR↗

arXiv2019

The Enskog process for hard and soft potentials

The density of a moderately dense gas evolving in a vacuum is given by the solution of an Enskog equation. Recently we have constructed in [ARS17] the stochastic process that corresponds to the Enskog equation under suitable conditions. The Enskog process is identified as the solution of a McKean-Vlasov equation driven by a Poisson random measure. In this work, we continue the study for a wider class of collision kernels that includes hard and soft potentials. Based on a suitable particle approximation of binary collisions, the existence of an Enskog process is established.

math-ph↗

arXiv2019

Exponential ergodicity for stochastic equations of nonnegative processes with jumps

In this work, we study ergodicity of continuous time Markov processes on state space $\mathbb{R}_{\geq 0} := [0,\infty)$ obtained as unique strong solutions to stochastic equations with jumps. Our first main result establishes exponential ergodicity in the Wasserstein distance, provided the stochastic equation satisfies a comparison principle and the drift is dissipative. In particular, it is applicable to continuous-state branching processes with immigration (shorted as CBI processes), possibly with nonlinear branching mechanisms or in Lévy random environments. Our second main result establishes exponential ergodicity in total variation distance for subcritical CBI processes under a first moment condition on the jump measure for branching and a $\log$-moment condition on the jump measure for immigration.

math.PR↗

arXiv2019

Boundary behavior of multi-type continuous-state branching processes with immigration

In this article we provide a sufficient condition for a continuous-state branching process with immigration (CBI process) to not hit its boundary, i.e. for non-extinction. Our result applies to arbitrary dimension $d \geq 1$ and is formulated in terms of an integrability condition for its immigration and branching mechanisms $F$ and $R$. The proof is based on a suitable comparison with one-dimensional CBI processes and an existing result for one-dimensional CBI processes. The same technique is also used to provide a sufficient condition for transience of multi-type CBI processes.

math.PR↗

arXiv2019

Weak-coupling limit for ergodic environments

The main aim of this work is to establish an averaging principle for a wide class of interacting particle systems in the continuum. This principle is an important step in the analysis of Markov evolutions and is usually applied for the associated semigroups related to backward Kolmogorov equations, c.f. [Kur73]. Our approach is based on the study of forward Kolmogorov equations (a.k.a. Fokker- Planck equations). We describe a system evolving as a Markov process on the space of finite configurations, whereas its rates depend on the actual state of another (equilibrium) process on the space of locally finite configurations. We will show that ergodicity of the environment process implies the averaging principle for the solutions of the coupled Fokker-Planck equations.

math-ph↗

arXiv2018

Linear evolution equations in scales of Banach spaces

This work is devoted to the study of a class of linear time-inhomogeneous evolution equations in a scale of Banach spaces. Existence, uniquenss and stability for classical solutions is provided. We study also the associated dual Cauchy problem for which we prove uniqueness in the dual scale of Banach spaces. The results are applied to an infinite system of ordinary differential equations but also to the Fokker-Planck equation associated with the spatial logistic model in the continuum.

math.FA↗

arXiv2018

Existence of densities for stochastic differential equations driven by Lévy processes with anisotropic jumps

We study existence of densities for solutions to stochastic differential equations with Hölder continuous coefficients and driven by a $d$-dimensional Lévy process $Z=(Z_{t})_{t\geq 0}$, where, for $t>0$, the density function $f_{t}$ of $Z_{t}$ exists and satisfies, for some $(α_{i})_{i=1,\dots,d}\subset(0,2)$ and $C>0$, \begin{align*} \limsup\limits _{t \to 0}t^{1/α_{i}}\int\limits _{\mathbb{R}^{d}}|f_{t}(z+e_{i}h)-f_{t}(z)|dz\leq C|h|,\ \ h\in \mathbb{R},\ \ i=1,\dots,d. \end{align*} Here $e_{1},\dots,e_{d}$ denote the canonical basis vectors in $\mathbb{R}^{d}$. The latter condition covers anisotropic $(α_{1},\dots,α_{d})$-stable laws but also particular cases of subordinate Brownian motion. To prove our result we use some ideas taken from \citep{DF13}.

math.PR↗

arXiv2018

Existence of densities for multi-type CBI processes

Let X be a multi-type continuous-state branching process with immigration (CBI process) on state space $\mathbb{R}^d$. Denote by $g_t$, $t \geq 0$, the law of $X_{t}$. We provide sufficient conditions under which $g_t$ has, for each $t > 0$, a density with respect to the Lebesgue measure. Such density has, by construction, some anisotropic Besov regularity. Our approach neither relies on the use of Malliavin calculus nor on the study of corresponding Laplace transform.

math.PR↗

arXiv2018

On Stochastic Cucker-Smale flocking dynamics

We present a stochastic version of the Cucker-Smale flocking dynamics based on a markovian $N$-particle system of pair interactions with unbounded and, in general, non-Lipschitz continuous interaction potential. We establish the infinite particle limit $N \to \infty$ and identify the limit as a solution with a nonlinear martingale problem describing the law of a weak solution to a Vlasov-McKean stochastic equation with jumps. Moreover, we estimate the total variation and Wasserstein distance for the time-marginals from which uniqueness in the class of solutions having some finite exponential moments is deduced. Based on the uniqueness for the time-marginals we prove uniqueness in law for the Vlasov-McKean equation, i.e. we establish propagation of chaos.

math.PR↗

arXiv2017

Stochastic averaging for a spatial population model in random environment

In this work we study the non-equilibrium Markov state evolution for a spatial population model on the space of locally finite configurations $Γ^2 = Γ^+ \times Γ^-$ over $\mathbb{R}^d$ where particles are marked by spins $\pm$. Particles of type '+' reproduce themselves independently of each other and, moreover, die due to competition either among particles of the same type or particles of different type. Particles of type '-' evolve according to a non-equilibrium Glauber-type dynamics with activity $z$ and potential $ψ$. Let $L^S$ be the Markov operator for '+' -particles and $L^E$ the Markov operator for '-' -particles. The non-equilibrium state evolution $(μ_t^{\varepsilon})_{t \geq 0}$ is obtained from the Fokker-Planck equation with Markov operator $L^S + \frac{1}{\varepsilon}L^E$, $\varepsilon > 0$, which itself is studied in terms of correlation function evolution on a suitable chosen scale of Banach spaces. We prove that in the limiting regime $\varepsilon \to 0$ the state evolution $μ_t^{\varepsilon}$ converges weakly to some state evolution $\overlineμ_t$ associated to the Fokker-Planck equation with (heuristic) Markov operator obtained from $L^S$ by averaging the interactions of the system with the environment with respect to the unique invariant Gibbs measure of the environment.

math-ph↗

arXiv2017

Stochastic averaging principle for spatial Markov evolutions in the continuum

We study a spatial birth-and-death process on the phase space of locally finite configurations $Γ^+ \times Γ^-$ over $\mathbb{R}^d$. Dynamics is described by an non-equilibrium evolution of states obtained from the Fokker-Planck equation and associated with the Markov operator $L^+(γ^-) + \frac{1}{\varepsilon}L^-$, $\varepsilon > 0$. Here $L^-$ describes the environment process on $Γ^-$ and $L^+(γ^-)$ describes the system process on $Γ^+$, where $γ^-$ indicates that the corresponding birth-and-death rates depend on another locally finite configuration $γ^- \in Γ^-$. We prove that, for a certain class of birth-and-death rates, the corresponding Fokker-Planck equation is well-posed, i.e. there exists a unique evolution of states $μ_t^{\varepsilon}$ on $Γ^+ \times Γ^-$. Moreover, we give a sufficient condition such that the environment is ergodic with exponential rate. Let $μ_{\mathrm{inv}}$ be the invariant measure for the environment process on $Γ^-$. In the main part of this work we establish the stochastic averaging principle, i.e. we prove that the marginal of $μ_t^{\varepsilon}$ onto $Γ^+$ converges weakly to an evolution of states on $Γ^+$ associated with the averaged Markov birth-and-death operator $\overline{L} = \int_{Γ^-}L^+(γ^-)d μ_{\mathrm{inv}}(γ^-)$.

math-ph↗

arXiv2017

Evolution of states and mesoscopic scaling for two-component birth-and-death dynamics in continuum

Two coupled spatial birth-and-death Markov evolutions on $\mathbb{R}^d$ are obtained as unique weak solutions to the associated Fokker-Planck equations. Such solutions are constructed by its associated sequence of correlation functions satisfying the so-called Ruelle-bound. Using the general scheme of Vlasov scaling we are able to derive a system of non-linear, non-local mesoscopic equations describing the effective density of the particle system. The results are applied to several models of ecology and biology.

math.PR↗

arXiv2016

Non-autonomous interacting particle systems in continuum

A conservative Feller evolution on continuous bounded functions is constructed from a weakly continuous, time-inhomogeneous transition function describing a pure jump process on a locally compact Polish space. The transition function is assumed to satisfy a Foster-Lyapunov type condition. The results are applied to interacting particle systems in continuum, in particular to general birth-and-death processes (including jumps). Particular examples such as the BDLP and Dieckmann-Law model are considered in the end.

math.DS↗