arXiv research

Approximate inference algorithm is one of the fundamental research fields in machine learning. The two dominant theoretical inference frameworks in machine learning are variational inference (VI) and Markov chain Monte Carlo (MCMC). However, because of the fundamental limitation in the theory, it is very challenging to…

Study of Anosov flows using microlocal analysis for ergodicity and mixing properties.

New method recovers BSDE from financial data without ergodicity.

In the present paper we develop a framework in which questions of quantum ergodicity for operators acting on sections of hermitian vector bundles over Riemannian manifolds can be studied. We are particularly interested in the case of locally symmetric spaces. For locally symmetric spaces, we extend the recent construct…

The paper tackles learning to control systems with unknown parameters using Brownian noise.

The paper develops a model for sovereign debt dynamics with explicit maturity structure.

Given a heterogeneous time-series sample, the objective is to find points in time (called change points) where the probability distribution generating the data has changed. The data are assumed to have been generated by arbitrary unknown stationary ergodic distributions. No modelling, independence or mixing assumptions…

The paper studies harmonic map heat flow to flat tori, proving ergodic behavior and convergence to hyperbolic measure.

We propose a general framework for constructing and describing infinite type flat surfaces of finite area. Using this method, we characterize the range of dynamical behaviors possible for the vertical translation flows on such flat surfaces. We prove a sufficient condition for ergodicity of this flow and apply the cond…

EGFs use ergodicity to simplify generative flows for easier training and imitation learning.

Study counts ergodic measures in surface lamination strata.

With their origin in thermodynamics and symbolic dynamics, Gibbs measures are crucial tools to study the ergodic theory of the geodesic flow on negatively curved manifolds. We develop a framework (through Patterson-Sullivan densities) allowing us to get rid of compactness assumptions on the manifold, and prove many exi…

Extends magnetic flow theory results to higher dimensions.

This paper analyzes the convergence of dynamic HMC and NUTS methods.

New progress on frame flow ergodicity for nearly pinched manifolds.

Recent results on ergodic theory for Riemann surface laminations and foliations.

Formula connects foliated simplicial volume with group cost.

In this note we show that the Riemann moduli spaces $M_{g, n}$ equipped with the Weil--Petersson metric are quantum ergodic for $3g+n \geq 4$. We also provide other examples of singular spaces with ergodic geodesic flow for which quantum ergodicity holds.

We extend to orbifolds classical results on quantum ergodicity due to Shnirelman, Colin de Verdière and Zelditch, proving that, for any positive, first-order self-adjoint elliptic pseudodifferential operator P on a compact orbifold X with positive principal symbol p, ergodicity of the Hamiltonian flow of p implies quan…

Develops Patterson-Sullivan theory for coarse cocycles.

Strong stability of ergodic iterations proven without ergodic driving sequence.

The paper studies the ergodicity of frame flow on even-dimensional manifolds.

Non-ergodic measures found in horocycle flow on Abelian differentials.

Log-ergodic model improves velocity of money prediction.

'Ergodicity economics' is criticized as pseudoscience.

We construct an example of a uniquely ergodic measured foliation on a surface such that the associated translation flow on the orientation double cover is minimal but not uniquely ergodic. We then prove a geometric criterion for the horizontal foliation of a quadratic differential to be uniquely ergodic. The second the…

New method stabilizes quantum ergodicity for mixed quantization and partial hyperbolicity.

Let $X$ be a Hadamard manifold, and $Γ$ a non-elementary discrete group of isometries of $X$ which contains a rank one isometry. We relate the ergodic theory of the geodesic flow of the quotient orbifold $M=X/Γ$ to the behavior of the Poincar{é} series of $Γ$. Precisely, the aim of this paper is to extend the so-called…

The study shows that ergodic measures are not generic on non-positively curved manifolds.

We show that Masur's logarithmic law of geodesics in the moduli space of translation surfaces does not imply unique ergodicity of the translation flow, but that a similar law involving the flat systole of a Teichmüller geodesic does imply unique ergodicity. It shows that the flat geometry has a better control on ergodi…

We relate ergodic-theoretic properties of a very small tree or lamination to the behavior of folding and unfolding paths in Outer space that approximate it, and we obtain a criterion for unique ergodicity in both cases. Our main result is that non-unique ergodicity gives rise to a transverse decomposition of the foldin…

The paper studies ergodicity of flows on subspaces, generalizing earlier work.

We present a general Markovian framework for order book modeling. Through our approach, we aim at providing a tool enabling to get a better understanding of the price formation process and of the link between microscopic and macroscopic features of financial assets. To do so, we propose a new method of order book repre…

A smooth diffeomorphism is said to be distributionally uniquely ergodic (DUE for short) when it is uniquely ergodic and its unique invariant probability measure is the only invariant distribution (up to multiplication by a constant). Ergodic translations on tori are classical examples of DUE diffeomorphisms. In this ar…

A measured solenoid is a laminated space endowed with a tranversal measure invariant by holonomy, as defined in arXiv:0910.2836. A measured solenoid immersed in a smooth manifold produces a closed current (known as generalized Ruelle-Sullivan current). Uniquely ergodic solenoids are those for which there is a unique (u…

Proposes a method to estimate SDE noise from a single trajectory.

The paper connects function theory, dynamics, and ergodic theory via Thurston's theory.

Study on Wasserstein distance for numerical approximations of stochastic differential equations.

The paper introduces a new method to create stable ergodic actions on higher-dimensional manifolds.

This work investigates a mixture of LMC and RMHMC with MMALA for geometric ergodicity.

Study shows non-wandering, partially hyperbolic systems are ergodic.

We introduce the concept of solenoid as an abstract laminated space. We do a thorough study of solenoids, leading to the notion of ergodic and uniquely ergodic solenoids. We define generalized currents associated with immersions of oriented solenoids with a transversal measure into smooth manifolds, generalizing Ruelle…

Study shows mapping class group action is ergodic on specific representations.

We study how resetting affects geometric Brownian motion, showing it becomes stationary but remains non-ergodic.

Study approximates top Lyapunov exponents for surface mapping classes.

The paper provides a link between ergodic theory and symplectic topology. A classical notion of ergodic theory is a skew product map associated with a loop in a group of transformations. We study skew products which come from loops in the group of Hamiltonian diffeomorphisms of a symplectic manifold. Our main question …

ISALT uses inference to simulate SDEs with large time-steps, improving efficiency.

We review some developments on clustering stochastic processes and come with the conclusion that asymptotically consistent clustering algorithms can be obtained when the processes are ergodic and the dissimilarity measure satisfies the triangle inequality. Examples are provided when the processes are distribution ergod…