Hypocoercivity for Kolmogorov backward evolution equations and applications

Grothaus, Martin; Stilgenbauer, Patrik

doi:10.1016/j.jfa.2014.08.019

Cited by 50 publications

(140 citation statements)

References 42 publications

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“…Note that we basically just have to check these conditions in applications, since the machinery works in a quite abstract Hilbert space setting. [7] In particular, Grothaus and Stilgenbauer [7] contains the required domain issues and furthermore, conditions for proving hypocoercivity need to be verified only on a fixed operator core of the evolution operator. A rigorous formulation of the method from Dolbeault et al [13] is given in Grothaus and Stilgenbauer.…”

Section: Hypocoercivity Methodsmentioning

confidence: 99%

“…[7] In particular, Grothaus and Stilgenbauer [7] contains the required domain issues and furthermore, conditions for proving hypocoercivity need to be verified only on a fixed operator core of the evolution operator. By contrast, in Grothaus and Stilgenbauer [7] the "geometric" framework of the Langevin equation is discussed whose state space is the product manifold R d × S d−1 , that is, the fibre lay-down equation. For the "flat" case meaning just a Langevin equation without the assumption of spherical velocities the technique is elaborated in Grothaus and Stilgenbauer [16] in full detail.…”

Section: Hypocoercivity Methodsmentioning

confidence: 99%

“…As suggested in Klar et al, [6,7] we propose the smooth fibre lay-down model Informally speaking, one could let the (Gaussian) noise act on the acceleration, then integrating once smoothes the noise acting on the velocity-to be more accurate, we replace the spherical Brownian motion in (1) by a spherical Langevin process with vanishing potential.…”

Section: Smooth Modelmentioning

confidence: 99%

“…Furthermore, the interested reader might also have a look at Grothaus and Stilgenbauer [7,Lemmas 3.8,3.9] where the remaining assertions except for (D2) are checked.…”

Section: Remark 35 Conditions (D) For the Fibre Lay-down Equationmentioning

confidence: 99%

“…It is well-known that (D3) can be proven via hypoellipticity techniques for smooth potentials, see [20,Proposition 5.5] . Indeed, Grothaus and Stilgenbauer [7,Section 4] is devoted to this topic entirely. This in turn is linked to (D2) via the Lumer-Phillips theorem.…”

Section: Remark 35 Conditions (D) For the Fibre Lay-down Equationmentioning

confidence: 99%

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Hypocoercivity for geometric Langevin equations motivated by fibre lay‐down models arising in industrial application

2018

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In this article we review a powerful and fairly simple hypocoercivity method and its application to qualitative analysis of industrially relevant fibre lay‐down models. The hypocoercivity strategy is formulated in an abstract Hilbert space setting with all necessary conditions; we provide some interpretation of these conditions and briefly explain why they are needed to make the machinery working. Once the conditions are fulfilled we gain exponential decay of the strongly continuous semigroup associated to the fibre lay‐down process at an explicitly known rate. We interpret this result as describing for instance homogeneity of nonwoven fibre webs. Primarily, this article portrays the concepts, however it is based on original results by Grothaus and Stilgenbauer, which can be found in the previous studies.

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Section: Hypocoercivity Methodsmentioning

confidence: 99%

Section: Hypocoercivity Methodsmentioning

confidence: 99%

Section: Smooth Modelmentioning

confidence: 99%

“…Furthermore, the interested reader might also have a look at Grothaus and Stilgenbauer [7,Lemmas 3.8,3.9] where the remaining assertions except for (D2) are checked.…”

Section: Remark 35 Conditions (D) For the Fibre Lay-down Equationmentioning

confidence: 99%

Section: Remark 35 Conditions (D) For the Fibre Lay-down Equationmentioning

confidence: 99%

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Hypocoercivity for geometric Langevin equations motivated by fibre lay‐down models arising in industrial application

2018

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Ergodicity for neutral type SDEs with infinite length of memory

Bao

wang

Chen

2020

Mathematische Nachrichten

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A hypocoercivity related ergodicity method for Kolmogorov equations

Grothaus

Stilgenbauer

2014

Proc Appl Math and Mech

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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 the degenerate N -particle Langevin dynamics. IntroductionStudying the decay to equilibrium of degenerate kinetic equations is still an active and demanding mathematical research area lying in between modern Stochastics and Functional Analysis. Especially in the last decade, many results concerning the relaxation to equilibrium of the degenerate Langevin dynamics have been obtained in case the underlying potential is sufficiently smooth and nonsingular; the required mathematical tools e.g. rely on hypoellipticity, hypocoercivity or stochastic Lyapunov type techniques. However, in Statistical Mechanics the underlying potential in the Langevin dynamics is usually of Lennard-Jones type. Thus it is singular and, as far as we know, analyzing the decay to equilibrium in this singuar situation is not possible with the abovementioned methods. Nevertheless, suitable tools to handle this situation are provided in article [1] in which ergodicity with rate of convergence for the N -particle Langevin dynamics including singular interaction potentials is established. The method used in [1] is itself based on the strategy from [3]. Hence it is of interest to generalize the method from [1] and [3] to an abstract setting which is done below. We present a new abstract method in a Hilbert space framework which is suitable for proving ergodicity with explicit computable rate of convergence for various diffusion processes associated with a degenerate non-coercive Kolmogorov evolution operator L. Again the generator may have singular and nonsmooth coefficients. We mention that our newly developed method is based on an interplay between Functional Analysis (using operator semigroups) and Stochastics and relies on martingale methods. It is further interesting to note that we require conditions in the abstract setting similar to the assumptions made in an abstract hypocoercivity setting in [2] (and see [4]).The underlying proceeding summarizes the abstract method obtained in our article [5] (or see [6, Ch. 3]). We do not present any proofs. For all details, the reader is referred to the mentioned reference. Therein, also applications of the abstract method to the Langevin dynamics and the generalized fiber lay-down process arising in industrial mathematics are presented.

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Hypocoercivity for Kolmogorov backward evolution equations and applications

Cited by 50 publications

References 42 publications

Hypocoercivity for geometric Langevin equations motivated by fibre lay‐down models arising in industrial application

Hypocoercivity for geometric Langevin equations motivated by fibre lay‐down models arising in industrial application

Ergodicity for neutral type SDEs with infinite length of memory

A hypocoercivity related ergodicity method for Kolmogorov equations

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