ICML 2025

Laplace Transform Based Low-Complexity Learning
of Continuous Markov Semigroups
Vladimir R. Kostic * 1 2 Karim Lounici * 3 H´el`ene Halconruy * 4 5
Timoth´ee Devergne 1 Pietro Novelli 1 Massimiliano Pontil 1 6
Abstract
Markov processes serve as universal models for
many real-world random processes. This paper
presents a data-driven approach to learning these
models through the spectral decomposition of the
infinitesimal generator (IG) of the Markov semi-
group. Its unbounded nature complicates tradi-
tional methods such as vector-valued regression
and Hilbert-Schmidt operator analysis. Existing
techniques, including physics-informed kernel re-
gression, are computationally expensive and lim-
ited in scope, with no recovery guarantees for
transfer operator methods when the time-lag is
small. We propose a novel method leveraging the
IG’s resolvent, characterized by the Laplace trans-
form of transfer operators. This approach is robust
to time-lag variations, ensuring accurate eigen-
value learning even for small time-lags. Our statis-
tical analysis applies to a broader class of Markov
processes than current methods while reducing
computational complexity from quadratic to linear
in the state dimension. Finally, we demonstrate
our theoretical findings in several experiments.
1. Introduction
Markov semigroups play a critical role in modeling dynam-
ics of complex systems across various fields, including op-
tion pricing in finance (Karatzas & Shreve, 1991), molecular
dynamics (Sch¨utte & Huisinga, 2003) and climate modeling
(Majda et al., 2009), where understanding long-term behav-
ior is essential for accurate forecasting and interpretation.
*Equal contribution
1CSML, Istituto Italiano di Tecnologia,
Genova, Italy 2Faculty of Science, University of Novi Sad, Ser-
bia 3CMAP, ´Ecole Polytechnique, Paris, France 4SAMOVAR,
T´el´ecom Sud-Paris, Institut Polytechnique de Paris, France
5MODAL’X, Universit´e Paris Nanterre, France 6AI Center, Univer-
sity College London, UK. ...