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Jörg-Uwe Löbus

Publications and source records attributed to Jörg-Uwe Löbus.

11 recordsLinked to original sources
arXiv2024

Quasi-Invariance under Flows Generated by Non-Linear PDEs

The paper is concerned with the change of probability measures $μ$ along non-random probability measure valued trajectories $ν_t$, $t\in [-1,1]$. Typically solutions to non-linear PDEs, modeling spatial development as time progresses, generate such trajectories. Depending on in which direction the map $ν\equivν_0\mapstoν_t$ does not exit the state space, for $t\in [-1,0]$ or for $t\in [0,1]$, quasi-invariance of the measure $μ$ under the map $ν\mapstoν_t$ is established and the Radon-Nikodym derivative of $μ\circν_t$ with respect to $μ$ is determined. It is also investigated how Fréchet differentiability of the solution map of the PDE can contribute to the existence of this Radon-Nikodym derivative. The first application is a certain Boltzmann type equation. Here the Fréchet derivative of the solution map is calculated explicitly and quasi-invariance is established. The second application is a PDE related to the asymptotic behavior of a Fleming-Viot type particle system. Here quasi-invariance is obtained and it is demonstrated how this result can be used in order to derive a corresponding integration by parts formula.

math.AP↗
arXiv2022

Knudsen Type Group for Time in $\mathbb{R}$ and Related Boltzmann Type Equations

We consider certain Boltzmann type equations on a bounded physical and a bounded velocity space under the presence of both, reflective as well as diffusive boundary conditions. We introduce conditions on the shape of the physical space and on the relation between the reflective and the diffusive part in the boundary conditions such that the associated Knudsen type semigroup can be extended to time $t\in\mathbb{R}$. Furthermore, we provide conditions under which there exists a unique global solution to a Boltzmann type equation for time $t\ge 0$ or for time $t\in [τ_0,\infty)$ for some $τ_0<0$ which is independent of the initial value at time 0. Depending on the collision kernel, $τ_0$ can be arbitrarily small.

math.AP↗
arXiv2018

Boundedness of the Stationary Solution to the Boltzmann Equation with Spatial Smearing, Diffusive Boundary Conditions, and Lions' Collision Kernel

We investigate the Boltzmann equation with spatial smearing, diffusive boundary conditions, and Lions' collision kernel. Both, the physical as well as the velocity space, are assumed to be bounded. Existence and uniqueness of a stationary solution, which is a probability density, has been demonstrated in [3] under a certain smallness assumption on the collision term. We prove that whenever there is a stationary solution then it is a.e. positively bounded from below and above.

math.AP↗
arXiv2016

Infinite dimensional Ornstein-Uhlenbeck processes with unbounded diffusion - Approximation, quadratic variation, and Itô formula

The paper studies a class of Ornstein-Uhlenbeck processes on the classical Wiener space. These processes are associated with a diffusion type Dirichlet form whose corresponding diffusion operator is unbounded in the Cameron-Martin space. It is shown that the distributions of certain finite dimensional Ornstein-Uhlenbeck processes converge weakly to the distribution of such an infinite dimensional Ornstein-Uhlenbeck process. For the infinite dimensional processes, the ordinary scalar quadratic variation is calculated. Moreover, relative to the stochastic calculus via regularization, the scalar as well as the tensor quadratic variation are derived. A related Itô formula is presented.

math.PR↗
arXiv2015

Mosco Type Convergence of Bilinear Forms and Weak Convergence of $n$-Particle Systems

It is well known that Mosco (type) convergence is a tool in order to verify weak convergence of finite dimensional distributions of sequences of stochastic processes. In the present paper we are concerned with the concept of Mosco type convergence for non-symmetric stochastic processes and, in particular, $n$-particle systems in order to establish relative compactness.

math.PR↗
arXiv2014

Absolute Continuity under Time Shift for Ornstein-Uhlenbeck type Processes with Delay or Anticipation

The paper is concerned with one-dimensional two-sided Ornstein-Uhlenbeck type processes with delay or anticipation. We prove existence and uniqueness requiring almost sure boundedness on the left half-axis in case of delay and almost sure boundedness on the right half-axis in case of anticipation. For those stochastic processes $(X,P_μ)$ we calculate the Radon-Nikodym density under time shift of trajectories, $P_μ(dX_{\cdot -t})/P_μ(dX)$, $t\in {\Bbb R}$.

math.PR↗
arXiv2014

Mosco Type Convergence and Weak Convergence for a Fleming-Viot type Particle System

We are concerned with Mosco type convergence for a non-symmetric $n$-particle Fleming-Viot system $\{X_1,\ldots,X_n\}$ in a bounded $d$-dimensional domain $D$ with smooth boundary. Moreover, we are interested in relative compactness of the $n$-particle processes. It turns out that integration by parts relative to the initial measure and the generator is the appropriate mathematical tool. For finitely many particles, such integration by parts is established by using probabilistic arguments. For the limiting infinite dimensional configuration we use a result from infinite dimensional non-gaussian calculus.

math.PR↗
arXiv2014

A Class of infinite dimensional stochastic Processes with unbounded Diffusion

The paper studies Dirichlet forms on the classical Wiener space and the Wiener space over non-compact complete Riemannian manifolds. The diffusion operator is almost everywhere an unbounded operator on the Cameron--Martin space. In particular, it is shown that under a class of changes of the reference measure, quasi-regularity of the form is preserved. We also show that under these changes of the reference measure, derivative and divergence are closable with certain closable inverses. We first treat the case of the classical Wiener space and then we transfer the results to the Wiener space over a Riemannian manifold.

math.PR↗
arXiv2013

Weak Convergence of $n$-Particle Systems Using Bilinear Forms

The paper is concerned with the weak convergence of $n$-particle processes to deterministic stationary paths as $n\to\infty$. A Mosco type convergence of a class of bilinear forms is introduced. The Mosco type convergence of bilinear forms results in a certain convergence of the resolvents of the $n$-particle systems. Based on this convergence a criterion in order to verify weak convergence of invariant measures is established. Under additional conditions weak convergence of stationary $n$-particle processes to stationary deterministic paths is proved. The method is applied to the particle approximation of a Ginzburg-Landau type diffusion. The present paper is in close relation to the paper Löbus (2011/2012). Different definitions of bilinear forms and versions of Mosco type convergence are introduced. Both papers demonstrate that the choice of the form and the type of convergence relates to the particular particle system.

math.PR↗
arXiv2013

Absolute Continuity under Time Shift of Trajectories and Related Stochastic Calculus

The paper is concerned with a class of two-sided stochastic processes of the form $X=W+A$. Here $W$ is a two-sided Brownian motion with random initial data at time zero and $A\equiv A(W)$ is a function of $W$. Elements of the related stochastic calculus are introduced. In particular, the calculus is adjusted to the case when $A$ is a jump process. Absolute continuity of $(X,P_{\sbnu})$ under time shift of trajectories is investigated. For example under various conditions on the initial density with respect to the Lebesgue measure, $m=d\bnu/dx$, and on $A$ with $A_0=0$ we verify % {eqnarray*} \frac{P_{\sbnu}(dX_{\cdot -t})}{P_{\sbnu}(dX_\cdot)}=\frac{m(X_{-t})} {m(X_0)}\cdot\prod_i|\nabla_{W_0}X_{-t}|_i {eqnarray*} % a.e. where the product is taken over all coordinates. Here $\sum_i(\nabla_{W_0}X_{-t})_i$ is the divergence of $X_{-t}$ with respect to the initial position. Crucial for this is the {\it temporal homogeneity} in the sense that $X(W_{\cdot +v}+A_v\1)=X_{\cdot+v}(W)$, $v\in {\Bbb R}$, where $A_v\1$ is the trajectory taking the constant value $A_v (W)$. By means of such a density, partial integration relative to the generator of the process $X$ is established. Relative compactness of sequences of such processes is established.

math.PR↗