Large deviations for Markov processes with switching and homogenisation via Hamilton–Jacobi–Bellman equations

We consider the context of molecular motors modelled by a diffusion process driven by the gradient of a weakly periodic potential that depends on an internal degree of freedom. The switch of the internal state, that can freely be interpreted as a molecular switch, is modelled as a Markov jump process that depends on the location of the motor....

Hamilton–Jacobi equations for controlled gradient flows: The comparison principle

Motivated by recent developments in the fields of large deviations for interacting particle systems and mean field control, we establish a comparison principle for the Hamilton–Jacobi equation corresponding to linearly controlled gradient flows of an energy function E defined on a metric space (E,d). Our analysis is based on a systematic use...

A general convergence result for viscosity solutions of Hamilton-Jacobi equations and non-linear semigroups

We extend the Barles-Perthame procedure [4] (see also [22]) of semi-relaxed limits of viscosity solutions of Hamilton-Jacobi equations of the type f−λHf=h to the context of non-compact spaces. The convergence result allows for equations on a ‘converging sequence of spaces’ as well as Hamilton-equations written in terms of two equations in...

A hamilton-jacobi point of view on mean-field gibbs-non-gibbs transitions

We study the loss, recovery, and preservation of differentiability of time-dependent large deviation rate functions. This study is motivated by mean-field Gibbs-non-Gibbs transitions. The gradient of the rate-function evolves according to a Hamiltonian flow. This Hamiltonian flow is used to analyze the regularity of the time-dependent rate...

Flux large deviations of weakly interacting jump processes via well-posedness of an associated Hamilton-Jacobi equation

We establish uniqueness for a class of first-order Hamilton-Jacobi equations with Hamiltonians that arise from the large deviations of the empirical measure and empirical flux pair of weakly interacting Markov jump processes. As a corollary, we obtain such a large deviation principle in the context of weakly interacting processes with time...

The exponential resolvent of a markov process and large deviations for markov processes via hamilton-jacobi equations

We study the Hamilton-Jacobi equation f − λHf = h, where Hf = e^−f Ae^f and where A is an operator that corresponds to a well-posed martingale problem. We identify an operator that gives viscosity solutions to the Hamilton-Jacobi equa-tion, and which can therefore be interpreted as the resolvent of H. The operator is given...

Well-posedness of Hamilton–Jacobi equations in population dynamics and applications to large deviations

We prove Freidlin–Wentzell type large deviation principles for various rescaled models in populations dynamics that have immigration and possibly harvesting: birth–death processes, Galton–Watson trees, epidemic SI models, and prey–predator models. The proofs are carried out using a general analytic approach based on the well-posedness of a...

Path-space moderate deviations for a Curie-Weiss model of self-organized criticality

The dynamical Curie-Weiss model of self-organized criticality (SOC) was introduced in (Ann. Inst. Henri Poincaré Probab. Stat. 53 (2017) 658-678) and it is derived from the classical generalized Curie-Weiss by imposing a microscopic Markovian evolution having the distribution of the Curie-Weiss model of SOC (Ann. Probab. 44 (2016) 444-478) as...

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

We modify the spin-flip dynamics of the Curie–Weiss model with dissipation in Dai Pra, Fischer and 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...

Classical large deviation theorems on complete Riemannian manifolds

We generalize classical large deviation theorems to the setting of complete, smooth Riemannian manifolds. We prove the analogue of Mogulskii's theorem for geodesic random walks via a general approach using viscosity solutions for Hamilton–Jacobi equations. As a corollary, we also obtain the analogue of Cramér's theorem. The approach also...

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 moderate deviation principles via a general analytic approach based on convergence of nonlinear generators and...

Dynamical moderate deviations for the Curie-Weiss model

We derive moderate deviation principles for the trajectory of the empirical magnetization of the standard Curie–Weiss model via a general analytic approach based on convergence of generators and uniqueness of viscosity solutions for associated Hamilton–Jacobi equations. The moderate asymptotics depend crucially on the phase under consideration.

Large Deviations for Finite State Markov Jump Processes with Mean-Field Interaction Via the Comparison Principle for an Associated Hamilton–Jacobi Equation

We prove the large deviation principle (LDP) for the trajectory of a broad class of finite state mean-field interacting Markov jump processes via a general analytic approach based on viscosity solutions. Examples include generalized Ehrenfest models as well as Curie–Weiss spin flip dynamics with singular jump rates. The main step in the proof of...

Boundary singularities and characteristics of Hamilton–Jacobi equation

Boundary-value problems for first order PDEs are locally considered, when the classical sufficient condition for the solution existence does not hold, but a solution still exists, possibly defined on one or both sides of the boundary surface. We note three situations when such a surface (locally) arises: (1) the part of the boundary surface with...