---
_id: '607'
abstract:
- lang: eng
text: We study the Fokker-Planck equation derived in the large system limit of the
Markovian process describing the dynamics of quantitative traits. The Fokker-Planck
equation is posed on a bounded domain and its transport and diffusion coefficients
vanish on the domain's boundary. We first argue that, despite this degeneracy,
the standard no-flux boundary condition is valid. We derive the weak formulation
of the problem and prove the existence and uniqueness of its solutions by constructing
the corresponding contraction semigroup on a suitable function space. Then, we
prove that for the parameter regime with high enough mutation rate the problem
exhibits a positive spectral gap, which implies exponential convergence to equilibrium.Next,
we provide a simple derivation of the so-called Dynamic Maximum Entropy (DynMaxEnt)
method for approximation of observables (moments) of the Fokker-Planck solution,
which can be interpreted as a nonlinear Galerkin approximation. The limited applicability
of the DynMaxEnt method inspires us to introduce its modified version that is
valid for the whole range of admissible parameters. Finally, we present several
numerical experiments to demonstrate the performance of both the original and
modified DynMaxEnt methods. We observe that in the parameter regimes where both
methods are valid, the modified one exhibits slightly better approximation properties
compared to the original one.
acknowledgement: "JH and PM are funded by KAUST baseline funds and grant no. 1000000193
.\r\nWe thank Nicholas Barton (IST Austria) for his useful comments and suggestions.
\r\n\r\n"
article_processing_charge: No
author:
- first_name: Katarina
full_name: Bodova, Katarina
id: 2BA24EA0-F248-11E8-B48F-1D18A9856A87
last_name: Bodova
orcid: 0000-0002-7214-0171
- first_name: Jan
full_name: Haskovec, Jan
last_name: Haskovec
- first_name: Peter
full_name: Markowich, Peter
last_name: Markowich
citation:
ama: 'Bodova K, Haskovec J, Markowich P. Well posedness and maximum entropy approximation
for the dynamics of quantitative traits. Physica D: Nonlinear Phenomena.
2018;376-377:108-120. doi:10.1016/j.physd.2017.10.015'
apa: 'Bodova, K., Haskovec, J., & Markowich, P. (2018). Well posedness and maximum
entropy approximation for the dynamics of quantitative traits. Physica D: Nonlinear
Phenomena. Elsevier. https://doi.org/10.1016/j.physd.2017.10.015'
chicago: 'Bodova, Katarina, Jan Haskovec, and Peter Markowich. “Well Posedness and
Maximum Entropy Approximation for the Dynamics of Quantitative Traits.” Physica
D: Nonlinear Phenomena. Elsevier, 2018. https://doi.org/10.1016/j.physd.2017.10.015.'
ieee: 'K. Bodova, J. Haskovec, and P. Markowich, “Well posedness and maximum entropy
approximation for the dynamics of quantitative traits,” Physica D: Nonlinear
Phenomena, vol. 376–377. Elsevier, pp. 108–120, 2018.'
ista: 'Bodova K, Haskovec J, Markowich P. 2018. Well posedness and maximum entropy
approximation for the dynamics of quantitative traits. Physica D: Nonlinear Phenomena.
376–377, 108–120.'
mla: 'Bodova, Katarina, et al. “Well Posedness and Maximum Entropy Approximation
for the Dynamics of Quantitative Traits.” Physica D: Nonlinear Phenomena,
vol. 376–377, Elsevier, 2018, pp. 108–20, doi:10.1016/j.physd.2017.10.015.'
short: 'K. Bodova, J. Haskovec, P. Markowich, Physica D: Nonlinear Phenomena 376–377
(2018) 108–120.'
date_created: 2018-12-11T11:47:28Z
date_published: 2018-08-01T00:00:00Z
date_updated: 2023-09-19T10:38:34Z
day: '01'
department:
- _id: NiBa
- _id: GaTk
doi: 10.1016/j.physd.2017.10.015
external_id:
arxiv:
- '1704.08757'
isi:
- '000437962900012'
isi: 1
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/1704.08757
month: '08'
oa: 1
oa_version: Submitted Version
page: 108-120
publication: 'Physica D: Nonlinear Phenomena'
publication_status: published
publisher: Elsevier
publist_id: '7198'
quality_controlled: '1'
scopus_import: '1'
status: public
title: Well posedness and maximum entropy approximation for the dynamics of quantitative
traits
type: journal_article
user_id: c635000d-4b10-11ee-a964-aac5a93f6ac1
volume: 376-377
year: '2018'
...