arXiv subjects

Richard C. Kraaij

Publications and source records attributed to Richard C. Kraaij.

27 records · Page 2Linked to original sources

arXiv2019

Gamma convergence on path-spaces via convergence of viscosity solutions of Hamilton-Jacobi equations

We establish a framework that allows to prove Gamma-converge of functionals of Lagrangian form on spaces of trajectories based on convergence of viscosity solutions of associated Hamilton-Jacobi equations. Gamma convergence follows from a: equi-coercivity, b: Gamma convergence of the projected functional at time 0, c: convergence of the Hamiltonians that appear as Legendre transform of the Lagrangian in the path-space functional.

math.FA↗

arXiv2018

Path-space moderate deviations for a class of Curie-Weiss models with dissipation

We modify the Glauber dynamics of the Curie-Weiss model with dissipation in Dai Pra, Fischer, Regoli[2013] by considering arbitrary transition rates and we analyze the phase-portrait as well as the dynamics of moderate fluctuations for macroscopic observables. We obtain path-space moderate deviation principles via a general analytic approach based on the convergence of non-linear generators and uniqueness of viscosity solutions for associated Hamilton-Jacobi equations. The moderate asymptotics depend crucially on the phase we are considering and, moreover, their behavior may be influenced by the choice of the rates.

math.PR↗

arXiv2018

A generator approach to stochastic monotonicity and propagation of order

We study stochastic monotonicity and propagation of order for Markov processes with respect to stochastic integral orders characterized by cones of functions satisfying $Φf \geq 0$ for some linear operator $Φ$. We introduce a new functional analytic technique based on the generator $A$ of a Markov process and its resolvent. We show that the existence of an operator $B$ with positive resolvent such that $ΦA - B Φ$ is a positive operator for a large enough class of functions implies stochastic monotonicity. This establishes a technique for proving stochastic monotonicity and propagation of order that can be applied in a wide range of settings including various orders for diffusion processes with or without boundary conditions and orders for discrete interacting particle systems.

math.PR↗

arXiv2018

Classical large deviations theorems on complete Riemannian manifolds

We generalize classical large deviations theorems to the setting of complete Riemannian manifolds. We prove the analogue of Mogulskii's theorem for geodesic random walks via a general approach using visocity solutions for Hamilton-Jacobi equations. As a corollary, we also obtain the analogue of Cramér's theorem. The approach also provides a new proof of Schilder's theorem. Additionally, we provide a proof of Schilder's theorem by using an embedding into Euclidean space, together with Freidlin-Wentzell theory.

math.PR↗

arXiv2018

Path-space moderate deviation principles for the random field Curie-Weiss model

We analyze the dynamics of moderate fluctuations for macroscopic observables of the random field Curie Weiss model (i.e., standard Curie-Weiss model embedded in a site dependent, i.i.d. random environment). We obtain path space large deviation principles via a general analytic approach based on convergence of non-linear generators and uniqueness of viscosity solutions for associated Hamilton Jacobi equations. The moderate asymptotics depend crucially on the phase we consider and moreover, the space-time scale range for which fluctuations can be proven is restricted by the addition of the disorder.

math.PR↗

arXiv2017

Fluctuation symmetry leads to GENERIC equations with non-quadratic dissipation

We develop a formalism to discuss the properties of GENERIC systems in terms of corresponding Hamiltonians that appear in the characterization of large-deviation limits. We demonstrate how the GENERIC structure naturally arises from a certain symmetry in the Hamiltonian, which extends earlier work that has connected the large-deviation behaviour of reversible stochastic processes to the gradient-flow structure of their deterministic limit. Natural examples of application include particle systems with inertia.

math.AP↗

arXiv2017

Gibbs-non-Gibbs transition in the fuzzy Potts models with a Kac-type interaction: Closing the Ising gap

We complete the investigation of the Gibbs properties of the fuzzy Potts model on the d-dimensional torus with Kac interaction which was started by Jahnel and one of the authors. As our main result of the present paper, we extend the previous sharpness result of mean-field bounds to cover all possible cases of fuzzy transformations, allowing also for the occurrence of Ising classes. The closing of this previously left open Ising-gap involves an analytical argument showing uniqueness of minimizing profiles for certain non-homogeneous conditional variational problems.

math.PR↗

arXiv2016

Large deviations of the trajectory of empirical distributions of Feller processes on locally compact spaces

We study the large deviation behaviour of the trajectories of empirical distributions of independent copies of time-homogeneous Feller processes on locally compact metric spaces. Under the condition that we can find a suitable core for the generator of the Feller process, we are able to define a notion of absolutely continuous trajectories of measures in terms of some topology on this core. Also, we define a Hamiltonian in terms of the linear generator and a Lagrangian as its Legendre transform. We prove the large deviation principle and show that the rate function can be decomposed as a rate function for the initial time and an integral over the Lagrangian, finite only for absolutely continuous trajectories of measures. We apply this result for diffusion and Lévy processes on R^d, for pure jump processes with bounded jump kernel on arbitrary locally compact spaces and for discrete interacting particle systems. For diffusion processes, the theorem partly extends the Dawson and Gärtner theorem for non-interacting copies in the sense that it only holds for time-homogeneous processes, but on the other hand it holds for processes with degenerate diffusion matrix.

math.FA↗

arXiv2016

A large deviation perspective on exponential decay of entropy and lower bounds on the Ricci-curvature

We offer a new point of view on the (Modified) Log-Sobolev inequality and lower bounds on the Ricci-curvature in the setting where the dynamics are obtained as the limit of Markov processes. In this setting, the large deviation rate function of the stationary measures of the Markov processes, plays the role of entropy. We define an entropy-information inequality (EII) that generalizes the (MLSI) and is equivalent to exponential decay of the rate function along the flow, and define an entropy-convexity inequality (ECI) that serves as an analogue of a lower bound on the Ricci-curvature in this setting.

math.PR↗