Electronic Journal of Probability

Hypoelliptic multiscale Langevin diffusions: large deviations, invariant measures and small mass asymptotics

Wenqing Hu and Konstantinos Spiliopoulos

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Abstract

We consider a general class of hypoelliptic Langevin diffusions and study two related questions. The first question is large deviations for hypoelliptic multiscale diffusions as the noise and the scale separation parameter go to zero. The second question is small mass asymptotics of (a) the invariant measure corresponding to the hypoelliptic Langevin operator and of (b) related hypoelliptic Poisson equations. The invariant measure corresponding to the hypoelliptic problem and appropriate hypoelliptic Poisson equations enter the large deviations rate function due to the multiscale effects. Based on the small mass asymptotics we derive that the large deviations behavior of the multiscale hypoelliptic diffusion is consistent with the large deviations behavior of its overdamped counterpart. Additionally, we rigorously obtain an asymptotic expansion of the solution to relevant hypoelliptic Poisson equations with respect to the mass parameter, characterizing the order of convergence as the mass parameter goes to zero. The proof of convergence of invariant measures is of independent interest, as it involves an improvement of the hypocoercivity result for the kinetic Fokker-Planck equation. We do not restrict attention to gradient drifts and our proof provides explicit information on the dependence of the bounds of interest in terms of the mass parameter.

Article information

Source
Electron. J. Probab., Volume 22 (2017), paper no. 55, 38 pp.

Dates
Received: 7 April 2016
Accepted: 26 May 2017
First available in Project Euclid: 30 June 2017

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1498809677

Digital Object Identifier
doi:10.1214/17-EJP72

Mathematical Reviews number (MathSciNet)
MR3672831

Zentralblatt MATH identifier
1368.60027

Subjects
Primary: 60F10: Large deviations 60G99: None of the above, but in this section 60H10: Stochastic ordinary differential equations [See also 34F05] 35H10: Hypoelliptic equations

Keywords
large deviations hypoelliptic multiscale diffusions homogenization hypocoercivity non-gradient systems

Rights
Creative Commons Attribution 4.0 International License.

Citation

Hu, Wenqing; Spiliopoulos, Konstantinos. Hypoelliptic multiscale Langevin diffusions: large deviations, invariant measures and small mass asymptotics. Electron. J. Probab. 22 (2017), paper no. 55, 38 pp. doi:10.1214/17-EJP72. https://projecteuclid.org/euclid.ejp/1498809677


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