Abstract
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.
Similar content being viewed by others
References
R. Adams and J. Fournier. Sobolev spaces. Academic Press, Amsterdam, 2. ed., repr. edition, 2008.
L. Beznea, N. Boboc, and M. Röckner. Markov processes associated with \(L^p\)-resolvents and applications to stochastic differential equations on Hilbert space. J. Evol. Equ., 6(4):745–772, 2006.
D. Cohn. Measure theory. Birkhaeuser, Boston, 1980.
F. Conrad. Construction and analysis of langevin dynamics in continuous particle systems. PhD thesis, Technische Universität Kaiserslautern, 2010.
F. Conrad and M. Grothaus. Construction of \(N\)-particle langevin dynamics for \(H^{1,\infty }\)-potentials via generalized Dirichlet forms. Potential Anal., 28(3):261–282, 2008.
F. Conrad and M. Grothaus. Construction, ergodicity and rate of convergence of \(N\)-particle Langevin dynamics with singular potentials. Journal of Evolution Equations, 10(3):623–662, 2010.
E. Davies. One-parameter semigroups. Academic Press, London, 1980.
A. Eberle. Uniqueness and Non-Uniqueness of Semigroups Generated by Singular Diffusion Operators. Springer, Heidelberg, 1999.
S. Ethier and T. Kurtz. Markov processes characterization and convergence. Wiley, New York, 1986.
L. Evans. Partial Differential Equations. American Mathematical Society, Providence, 2010.
M. Freidlin. Some remarks on the Smoluchowski-Kramers approximation. J. Statist. Phys., 117(3-4):617–634, 2004.
J. Goldstein. Semigroups of linear operators and applications. Oxford University Press, New York, 1985.
M. Grothaus and P. Stilgenbauer. A hypocoercivity related ergodicity method for singularly distorted non-symmetric diffusions. Integral equations and operator theory, 83(3):331–379, 2015.
M. Grothaus and F. Wang. Weak Poincaré Inequalities for Convergence Rate of Degenerate Diffusion Processes. Ann. Probab. (to appear), 2019. arXiv:1703.04821
D. Herzog, S. Hottovy, and G. Volpe. The small-mass limit for Langevin dynamics with unbounded coefficients and positive friction. J. Stat. Phys., 163(3):659–673, 2016.
S. Hottovy, A. McDaniel, G. Volpe, and J. Wehr. The Smoluchowski-Kramers limit of stochastic differential equations with arbitrary state-dependent friction. Comm. Math. Phys., 336(3):1259–1283, 2015.
I. Karatzas and S. Shreve. Brownian motion and stochastic calculus. Springer, New York, springer study ed., 2. ed., corr. 8. print. edition, 2005.
N.V. Krylov and Michael Röckner. Strong solutions of stochastic equations with singular time dependent drift. Probability Theory and Related Fields, 131(2):154–196, 2005.
K. Kuwae and T. Shioya. Convergence of spectral structures: A functional analytic theory and its applications to spectral geometry. Communications in analysis and geometry, 11(4):599–674, 2003.
Z. Ma and M. Röckner. Introduction to the theory of (non-symmetric) Dirichlet forms. Springer, Berlin, 1992.
V. Nabiullin. Convergence of the langevin dynamics to a distorted Brownian motion in the small velocity limit - an operator semigroup approach. Masters Thesis, Technische Universität Kaiserslautern, 2014.
G. Pavliotis. Stochastic Processes and Applications Diffusion Processes, the Fokker-Planck and Langevin Equations. Springer, Heidelberg, 2014.
M. Rousset, Y. Xu, and P. Zitt. A weak overdamped limit theorem for langevin processes. 2017. arXiv:1709.09866
J. Tölle. Convergence of non-symmetric forms with changing reference measures. 2006. https://bibos.math.uni-bielefeld.de/preprints/E06-09-234.pdf
Acknowledgements
The second author thanks the department of Mathematics at the University of Kaiserslautern for financial support in the form of a fellowship.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Grothaus, M., Nonnenmacher, A. Overdamped limit of generalized stochastic Hamiltonian systems for singular interaction potentials. J. Evol. Equ. 20, 577–605 (2020). https://doi.org/10.1007/s00028-019-00530-8
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00028-019-00530-8
Keywords
- Markov semigroups
- Langevin equations
- Overdamped limit
- Distorted Brownian motion
- Semigroup convergence on varying spaces