105 research outputs found
Overdamped limit of generalized stochastic Hamiltonian systems for singular interaction potentials
First weak solutions of generalized stochastic Hamiltonian systems (gsHs) are
constructed via essential m-dissipativity of their generators on a suitable
core. For a scaled gsHs we prove convergence of the corresponding semigroups
and tightness of the weak solutions. This yields convergence in law of the
scaled gsHs to a distorted Brownian motion. In particular, the results confirm
the convergence of the Langevin dynamics in the overdamped regime to the
overdamped Langevin equation. The proofs work for a large class of (singular)
interaction potentials including, e.g., potentials of Lennard--Jones type
A hypocoercivity related ergodicity method with rate of convergence for singularly distorted degenerate Kolmogorov equations and applications
In this article we develop a new abstract strategy for proving ergodicity
with explicit computable rate of convergence for diffusions associated with a
degenerate Kolmogorov operator L. A crucial point is that the evolution
operator L may have singular and nonsmooth coefficients. This allows the
application of the method e.g. to degenerate and singular particle systems
arising in Mathematical Physics. As far as we know in such singular cases the
relaxation to equilibrium can't be discussed with the help of existing
approaches using hypoellipticity, hypocoercivity or stochastic Lyapunov type
techniques. The method is formulated in an L2-Hilbert space setting and is
based on an interplay between Functional Analysis and Stochastics. Moreover, it
implies an ergodicity rate which can be related to L2-exponential convergence
of the semigroup. Furthermore, the ergodicity method shows up an interesting
analogy with existing hypocoercivity approaches. In the first application we
discuss ergodicity of the N-particle degenerate Langevin dynamics with singular
potentials. The dual to this equation is also called the kinetic Fokker-Planck
equation with an external confining potential. In the second example we apply
the method to the so-called (degenerate) spherical velocity Langevin equation
which is also known as the fiber lay-down process arising in industrial
mathematics.Comment: Older preprint version of the paper (from 2014). The final
publication is available at Springer via
http://dx.doi.org/10.1007/s00020-015-2254-1 It was published under the name
"A hypocoercivity related ergodicity method for singularly distorted
non-symmetric diffusions"; see Integral Equations Oper. Theory 83, No. 3,
Article ID 2254, 331-379 (2015
Mittag-Leffler Analysis II: Application to the fractional heat equation
Mittag-Leffler analysis is an infinite dimensional analysis with respect to
non-Gaussian measures of Mittag-Leffler type which generalizes the powerful
theory of Gaussian analysis and in particular white noise analysis. In this
paper we further develop the Mittag-Leffler analysis by characterizing the
convergent sequences in the distribution space. Moreover we provide an
approximation of Donsker's delta by square integrable functions. Then we apply
the structures and techniques from Mittag-Leffler analysis in order to show
that a Green's function to the time-fractional heat equation can be constructed
using generalized grey Brownian motion (ggBm) by extending the fractional
Feynman-Kac formula from Schneider. Moreover we analyse ggBm, show its
differentiability in a distributional sense and the existence of corresponding
local times.Comment: 45 page
A Fundamental Solution to the Schr\"odinger Equation with Doss Potentials and its Smoothness
We construct a fundamental solution to the Schr\"odinger equation for a class
of potentials of polynomial type by a complex scaling approach as in
[Doss1980]. The solution is given as the generalized expectation of a white
noise distribution. Moreover, we obtain an explicit formula as the expectation
of a function of Brownian motion. This allows to show its differentiability in
the classical sense. The admissible potentials may grow super-quadratically,
thus by a result from [Yajima1996] the solution does not belong to the
self-adjoint extension of the Hamiltonian.Comment: 32 page
A White Noise Approach to Phase Space Feynman Path Integrals
The concepts of phase space Feynman integrals in White Noise Analysis are
established. As an example the harmonic oscillator is treated. The approach
perfectly reproduces the right physics. I.e., solutions to the Schr\"odinger
equation are obtained and the canonical commutation relations are satisfied.
The later can be shown, since we not only construct the integral but rather the
Feynman integrand and the corresponding generating functional
The Hamiltonian Path Integrand for the Charged Particle in a Constant Magnetic field as White Noise Distribution
The concepts of Hamiltonian Feynman integrals in white noise analysis are
used to realize as the first velocity dependent potential the Hamiltonian
Feynman integrand for a charged particle in a constant magnetic field in
coordinate space as a Hida distribution. For this purpose the velocity
dependent potential gives rise to a generalized Gauss kernel. Besides the
propagators also the generating functionals are obtained.Comment: arXiv admin note: substantial text overlap with arXiv:1203.0089,
arXiv:1012.112
Weak Poincar\'e Inequalities for Convergence Rate of Degenerate Diffusion Processes
For a contraction -semigroup on a separable Hilbert space, the decay
rate is estimated by using the weak Poincar\'e inequalities for the symmetric
and anti-symmetric part of the generator. As applications, non-exponential
convergence rate is characterized for a class of degenerate diffusion
processes, so that the study of hypocoercivity is extended. Concrete examples
are presented
An Invariance Principle for the Tagged Particle Process in Continuum with Singular Interaction Potential
We consider the dynamics of a tagged particle in an infinite particle
environment moving according to a stochastic gradient dynamics. For singular
interaction potentials this tagged particle dynamics was constructed first in
[FG11], using closures of pre-Dirichlet forms which were already proposed in
[GP87] and [Osa98]. The environment dynamics and the coupled dynamics of the
tagged particle and the environment were constructed separately. Here we
continue the analysis of these processes: Proving an essential m-dissipativity
result for the generator of the coupled dynamics from [FG11], we show that this
dynamics does not only contain the environment dynamics (as one component), but
is, given the latter, the only possible choice for being the coupled process.
Moreover, we identify the uniform motion of the environment as the reversed
motion of the tagged particle. (Since the dynamics are constructed as
martingale solutions on configuration space, this is not immediate.)
Furthermore, we prove ergodicity of the environment dynamics, whenever the
underlying reference measure is a pure phase of the system. Finally, we show
that these considerations are sufficient to apply [DMFGW89] for proving an
invariance principle for the tagged particle process. We remark that such an
invariance principle was studied before in [GP87] for smooth potentials, and
shown by abstract Dirichlet form methods in [Osa98] for singular potentials.
Our results apply for a general class of Ruelle measures corresponding to
potentials possibly having infinite range, a non-integrable singularity at 0
and a nontrivial negative part, and fulfill merely a weak differentiability
condition on .Comment: in comparison to version 1 slight changes in the introduction onl
Construction and analysis of a sticky reflected distorted Brownian motion
We give a Dirichlet form approach for the construction of a distorted
Brownian motion in , , where the behavior on
the boundary is determined by the competing effects of reflection from and
pinning at the boundary (sticky boundary behavior). In providing a Skorokhod
decomposition of the constructed process we are able to justify that the
stochastic process is solving the underlying stochastic differential equation
weakly for quasi every starting point with respect to the associated Dirichlet
form. That the boundary behavior of the constructed process indeed is sticky,
we obtain by proving ergodicity of the constructed process. Therefore, we are
able to show that the occupation time on specified parts of the boundary is
positive. In particular, our considerations enable us to construct a dynamical
wetting model (also known as Ginzburg--Landau dynamics) on a bounded set
under mild assumptions on the
underlying pair interaction potential in all dimensions . In
dimension this model describes the motion of an interface resulting from
wetting of a solid surface by a fluid.Comment: arXiv admin note: substantial text overlap with arXiv:1203.207
A White Noise Approach to the Feynman Integrand for Electrons in Random Media
Using the Feynman path integral representation of quantum mechanics it is
possible to derive a model of an electron in a random system containing dense
and weakly-coupled scatterers, see [Proc. Phys. Soc. 83, 495-496 (1964)]. The
main goal of this paper is to give a mathematically rigorous realization of the
corresponding Feynman integrand in dimension one based on the theory of white
noise analysis. We refine and apply a Wick formula for the product of a
square-integrable function with Donsker's delta functions and use a method of
complex scaling. As an essential part of the proof we also establish the
existence of the exponential of the self-intersection local times of a
one-dimensional Brownian bridge. As result we obtain a neat formula for the
propagator with identical start and end point. Thus, we obtain a well-defined
mathematical object which is used to calculate the density of states, see e.g.
[Proc. Phys. Soc. 83, 495-496 (1964)].Comment: final versio
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