arXiv research

We study a phenomenological model for the continuous double auction, equivalent to two independent $M/M/1$ queues. The continuous double auction defines a continuous-time random walk for trade prices. The conditions for ergodicity of the auction are derived and, as a consequence, three possible regimes in the behavior …

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

Using elements from the theory of ergodic backward stochastic differential equations (BSDE), we study the behavior of forward entropic risk measures. We provide their general representation results (via both BSDE and convex duality) and examine their behavior for risk positions of long maturities. We show that forward …

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

Log-ergodic model improves velocity of money prediction.

We introduce and solve a new type of quadratic backward stochastic differential equation systems defined in an infinite time horizon, called \emph{ergodic BSDE systems}. Such systems arise naturally as candidate solutions to characterize forward performance processes and their associated optimal trading strategies in a…

We describe a method for constructing Teichmüller geodesics where the vertical measured foliation $ν$ is minimal but is not uniquely ergodic and where we have a good understanding of the behavior of the Teichmüller geodesic. The construction depends on various parameters, and we show that one can adjust the parameters …

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…

Study long-term behavior of semi-Markov modulated processes using integral functions.

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…

In this paper we study the convergence behavior of grafting rays to the Thurston boundary of Teichmuller space. When the grafting is done along a weighted system of simple closed curves or along a maximal uniquely ergodic lamination this behavior is the same as for Teichmuller geodesics and lines of minima. We also sho…

Study of Moncrief lines' behavior in curved space-times.

The paper explores properties of functions on Teichmüller space, proving theorems about limits and non-ergodicity.

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…

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

This paper compares two NUTS variants and analyzes their convergence and mixing times.

We explore nature of price formation in financial markets and develop a theory of bid and ask price dynamics in which the two prices form due to quantum-chaotic interaction between buy and sell orders. In this model bid and ask prices are represented by eigenvalues of a 2x2 price operator corresponding to 'bid' and 'as…

We study asymptotic properties of maximum likelihood estimators for Heston models based on continuous time observations of the log-price process. We distinguish three cases: subcritical (also called ergodic), critical and supercritical. In the subcritical case, asymptotic normality is proved for all the parameters, whi…

Study magnetic Laplacians on hyperbolic surfaces, revealing three regimes of eigenfunction behavior.

Study counts ergodic measures in surface lamination strata.

The method of cointegration in regression analysis is based on an assumption of stationary increments. Stationary increments with fixed time lag are called integration I(d). A class of regression models where cointegration works was identified by Granger and yields the ergodic behavior required for equilibrium expectat…

Conditions for geometric ergodicity of multivariate autoregressive conditional heteroskedasticity (ARCH) processes, with the so-called BEKK (Baba, Engle, Kraft, and Kroner) parametrization, are considered. We show for a class of BEKK-ARCH processes that the invariant distribution is regularly varying. In order to accou…

This paper addresses metaconsistency in Bayesian inference for metastable systems.

Stochastic approximation algorithms show exponential progress bounds.

The Planck mass can be derived from gravitational potential behavior in compactifications.

AdaBoost is one of the most popular ML algorithms. It is simple to implement and often found very effective by practitioners, while still being mathematically elegant and theoretically sound. AdaBoost's interesting behavior in practice still puzzles the ML community. We address the algorithm's stability and establish m…

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…

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.

'Ergodicity economics' is criticized as pseudoscience.

The paper improves regret lower bounds for communicating MDPs.

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.

This survey reviews portfolio selection problem for long-term horizon. We consider two objectives: (i) maximize the probability for outperforming a target growth rate of wealth process (ii) minimize the probability of falling below a target growth rate. We study the asymptotic behavior of these criteria formulated as l…

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…

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

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…

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

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.