PreprintCC BY 4.0arXiv
Connection of hypocoercivity and hypocontractivity via the θ-methods
Abstract
Recent literature shows that hypocoercivity properties of linear evolution equations (in particular their exponential decay and the sharp short time decay of their propagator norm) carry over to their discretization via the midpoint rule. This note discusses this connection for the (other) θ-methods, i.e.\ for θ≠1/2. It is shown that any implicit discretization with θ∈ (1/2,1] (pertaining to a hypocoercive continuous-time evolution equation) is contractive, and not only hypocontractive -- in contrast to the midpoint rule. For a coercive continuous-time evolution equation, a discretization with θ∈ [0,1/2) is contractive for time steps small enough.
Related
- Hypocoercivity-preserving space-time Galerkin methods for kinetic Fokker-Planck equations
- Numerical Neural Operator: Connection Between Numerical Analysis and the Operator Learning and Building Discretized Neural Operators from Finite Element Methods
- Fractional hypocoercivity in bounded domains in the anomalous diffusion limit
- Hypocoercivity of Tempered Bouncy Particle Samplers for Heavy-Tailed Targets
- Weighted L2 bounds for kinetic Fokker-Planck equations and hypocoercivity
BibTeX
@misc{arnold2026connection,
title = {{Connection of hypocoercivity and hypocontractivity via the $θ$-methods}},
author = {Anton Arnold and Stefan Egger},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2605.30197},
eprint = {2605.30197},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.30197},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Connection of hypocoercivity and hypocontractivity via the θ-methods
AU - Anton Arnold
AU - Stefan Egger
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2605.30197
UR - https://arxiv.org/abs/2605.30197
AB - Recent literature shows that hypocoercivity properties of linear evolution equations (in particular their exponential decay and the sharp short time decay of their propagator norm) carry over to their discretization via the midpoint rule. This note discusses this connection for the (other) θ-methods, i.e.\ for θ≠1/2. It is shown that any implicit discretization with θ∈ (1/2,1] (pertaining to a hypocoercive continuous-time evolution equation) is contractive, and not only hypocontractive -- in contrast to the midpoint rule. For a coercive continuous-time evolution equation, a discretization with θ∈ [0,1/2) is contractive for time steps small enough.
ER -