arXiv subjects

Francis Lörler

Publications and source records attributed to Francis Lörler.

7 recordsLinked to original sources

arXiv2026

Relaxation times of non-reversible Markov processes

We develop a systematic approach to quantify $L^2$-relaxation times for non-reversible Markov processes based on the singular value gap of the generator introduced by Chatterjee. The inverse of the singular value gap is equivalent to the relaxation time of the time-averaged transition semigroup. We show that, moreover, the singular value gap of the two-point motion also provides a lower bound on the usual spectral gap of the generator, and its inverse provides upper bounds on relaxation times without time averaging for sufficiently regular initial laws. We then introduce a method for deriving lower bounds on singular value gaps for Markov processes with degenerate noise that is based on the concept of a first- and second-order collapse of the generator. It follows ideas from hypocoercivity developed in a previous series of works but is simpler and more broadly applicable. In contrast to previous results, it includes settings with non-vanishing first-order collapse, and thus applies directly to Markov chains (in continuous time), but also to diffusion processes and piecewise-deterministic Markov processes. Our approach yields sharp upper and lower bounds for several classes of examples. First applications include the proof of a conjecture by Diaconis and Miclo on a square-root speed-up for lifted random walks on abelian groups, as well as bounds on relaxation times of switching flows perturbed by noise and of non-reversible diffusion processes.

math.PR↗

arXiv2026

Convergence rates of self-repelling diffusions on Riemannian manifolds

We study a class of self-repelling diffusions on compact Riemannian manifolds whose drift is the gradient of a potential accumulated along their trajectory. When the interaction potential admits a suitable spectral decomposition, the dynamics and its environment are equivalent to a finite-dimensional degenerate diffusion. We show that this diffusion is a second-order lift of an Ornstein-Uhlenbeck process whose invariant law corresponds to the Gaussian invariant measure of the environment, and immediately obtain a general upper bound on the rate of convergence to stationarity using the framework of second-order lifts. Furthermore, using a flow Poincaré inequality, we develop lower bounds on the convergence rate. We show that, in the periodic case, these lower bounds improve upon those of Benaïm and Gauthier (Probab. Theory Relat. Fields, 2016), and even match the order of the upper bound in some cases.

math.PR↗

arXiv2025

Convergence rates of self-repellent random walks, their local time and Event Chain Monte Carlo

We study the rate of convergence to equilibrium of the self-repellent random walk and its local time process on the discrete circle $\mathbb{Z}_n$. While the self-repellent random walk alone is non-Markovian since the jump rates depend on its history via its local time, jointly considering the evolution of the local time profile and the position yields a piecewise deterministic, non-reversible Markov process. We show that this joint process can be interpreted as a second-order lift of a reversible diffusion process, the discrete stochastic heat equation with Gaussian invariant measure. In particular, we obtain a lower bound on the relaxation time of order $Ω(n^{3/2})$. Using a flow Poincaré inequality, we prove an upper bound for a slightly modified dynamics of order $O(n^2)$, matching recent conjectures in the physics literature. Furthermore, since the self-repellent random walk and its local time process coincide with the Event Chain Monte Carlo algorithm for the harmonic chain, a non-reversible MCMC method, we demonstrate that the relaxation time bound confirms the recent empirical observation that Event Chain Monte Carlo algorithms can outperform traditional MCMC methods such as Hamiltonian Monte Carlo.

math.PR↗

arXiv2025

Convergence of non-reversible Markov processes via lifting and flow Poincar{é} inequality

We propose a general approach for quantitative convergence analysis of non-reversible Markov processes, based on the concept of second-order lifts and a variational approach to hypocoercivity. To this end, we introduce the flow Poincar{é} inequality, a space-time Poincar{é} inequality along trajectories of the semigroup, and a general divergence lemma based only on the Dirichlet form of an underlying reversible diffusion. We demonstrate the versatility of our approach by applying it to a pair of run-and-tumble particles with jamming, a model from non-equilibrium statistical mechanics, and several piecewise deterministic Markov processes used in sampling applications, in particular including general stochastic jump kernels.

math.AP↗

arXiv2025

Hypocoercivity meets lifts

We unify the variational hypocoercivity framework established by D. Albritton, S. Armstrong, J.-C. Mourrat, and M. Novack, with the notion of second-order lifts of reversible diffusion processes, recently introduced by A. Eberle and F. Lörler. We give an abstract, yet fully constructive, presentation of the theory, so that it can be applied to a large class of linear kinetic equations. As this hypocoercivity technique does not twist the reference norm, we can recover accurate and sharp convergence rates in various models. Among those, adaptive Langevin dynamics (ALD) is discussed in full detail and we show that for near-quadratic potentials, with suitable choices of parameters, it is a near-optimal second-order lift of the overdamped Langevin dynamics. As a further consequence, we observe that the Generalised Langevin Equation (GLE) is a also a second-order lift, as the standard (kinetic) Langevin dynamics are, of the overdamped Langevin dynamics. Then, convergence of (GLE) cannot exceed ballistic speed, i.e. the square root of the rate of the overdamped regime. We illustrate this phenomenon with explicit computations in a benchmark Gaussian case.

math.PR↗

arXiv2024

Space-time divergence lemmas and optimal non-reversible lifts of diffusions on Riemannian manifolds with boundary

Non-reversible lifts reduce the relaxation time of reversible diffusions at most by a square root. For reversible diffusions on domains in Euclidean space, or, more generally, on a Riemannian manifold with boundary, non-reversible lifts are in particular given by the Hamiltonian flow on the tangent bundle, interspersed with random velocity refreshments, or perturbed by Ornstein-Uhlenbeck noise, and reflected at the boundary. In order to prove that for certain choices of parameters, these lifts achieve the optimal square-root reduction up to a constant factor, precise upper bounds on relaxation times are required. A key tool for deriving such bounds by space-time Poincaré inequalities is a quantitative space-time divergence lemma. Extending previous work of Cao, Lu and Wang, we establish such a divergence lemma with explicit constants for general locally convex domains with smooth boundary in Riemannian manifolds satisfying a lower, not necessarily positive, curvature bound. As a consequence, we prove optimality of the lifts described above up to a constant factor, provided the deterministic transport part of the dynamics and the noise are adequately balanced. Our results show for example that an integrated Ornstein-Uhlenbeck process on a locally convex domain with diameter $d$ achieves a relaxation time of the order $d$, whereas, in general, the Poincaré constant of the domain is of the order $d^2$.

math.PR↗

arXiv2024

Non-reversible lifts of reversible diffusion processes and relaxation times

We propose a new concept of lifts of reversible diffusion processes and show that various well-known non-reversible Markov processes arising in applications are lifts in this sense of simple reversible diffusions. Furthermore, we introduce a concept of non-asymptotic relaxation times and show that these can at most be reduced by a square root through lifting, generalising a related result in discrete time. Finally, we demonstrate how the recently developed approach to quantitative hypocoercivity based on space-time Poincaré inequalities can be rephrased and simplified in the language of lifts and how it can be applied to find optimal lifts.

math.PR↗