arXiv subjects

Richard C. Kraaij

Publications and source records attributed to Richard C. Kraaij.

At least 19 recordsLinked to original sources

arXiv2026

A Strict Comparison Principle for Integro-Differential Hamilton-Jacobi-Bellman Equations on Domains with Boundary

This work provides a comparison principle for viscosity solutions to boundary value problems on (partially) bounded, cylindrical spaces. The comparison principle is based on a test function framework, that allows for the simultaneous treatment of diffusive as well as jump terms. Estimates in the proof of the comparison principle incorporate the use of Lyapunov functions that act as growth bounds for the solutions, effectively yielding a theory for unbounded viscosity solutions. We apply the results to a wide class of parabolic equations and elliptic problems on a space with corners.

math.AP↗

arXiv2025

Existence of viscosity solutions for Hamilton-Jacobi equations via Lyapunov control

We give a new perspective on the existence of viscosity solutions for a stationary and a time-dependent first-order Hamilton-Jacobi equation. Following recent comparison principles, we work in a framework in which we consider a subsolution and a supersolution for two equations in terms of two Hamiltonians that can be seen as an upper semi-continuous upper and lower semi-continuous lower bound of our original Hamiltonian respectively. The bounds are made rigorous in terms of Youngs inequality. The bounds are furthermore formulated in a way that incorporate a Lyapunov function which allows us to restrict part of the analysis to compact sets and to work with almost optimizers of the considered control problems. For this reason, we can relax assumptions on the control problem: most notably, we do not need completeness of set of controlled paths. Moreover, this strategy avoids a-priori analysis on the regularity of the candidate solutions. To complete our picture, we exhibit our result in two contexts. First, we consider Riemannian manifolds with smooth boundary, in which the dynamics allows for both "inward" and "outward" drift. The boundary conditions are embedded into the Hamiltonians itself. Second, we consider the solutions to a Fillipov differential equation, i.e. one with discontinuous vector field. We show that our notion of Hamiltonians leverages the natural embedding of the discontinuity in an associated set-valued differential inclusion.

math.AP↗

arXiv2024

A comparison principle based on couplings of partial integro-differential operators

This paper is concerned with a comparison principle for viscosity solutions to Hamilton-Jacobi (HJ), -Bellman (HJB), and -Isaacs (HJI) equations for general classes of partial integro-differential operators. Our approach innovates in three ways: (1) We reinterpret the classical doubling-of-variables method in the context of second-order equations by casting the Ishii-Crandall Lemma into a test function framework. This adaptation allows us to effectively handle non-local integral operators, such as those associated with Lévy processes. (2) We translate the key estimate on the difference of Hamiltonians in terms of an adaptation of the probabilistic notion of couplings, providing a unified approach that applies to differential, difference, and integral operators. (3) We strengthen the sup-norm contractivity resulting from the comparison principle to one that encodes continuity in the strict topology. We apply our theory to a variety of examples, in particular, to second-order differential operators and, more generally, generators of spatially inhomogeneous Lévy processes.

math.AP↗

arXiv2024

Large deviations for slow-fast processes on connected complete Riemannian manifolds

We consider a class of slow-fast processes on a connected complete Riemannian manifold $M$.The limiting dynamics as the scale separation goes to $\infty$ is governed by the averaging principle. Around this limit, we prove large deviation principles with an action-integral rate function for the slow process by nonlinear semigroup methods together with the Hamilton-Jacobi-Bellman equation techniques. The innovation is solving a comparison principle for viscosity solutions on $M$ and the existence of a viscosity solution via a control problem for a non-smooth Hamiltonian.

math.PR↗

arXiv2024

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. Rescaling space and time, the limit of the trajectory of the diffusion process homogenizes over the periodic potential as well as over the internal degree of freedom. Around the homogenized limit, we prove the large deviation principle of trajectories with a method developed by Feng and Kurtz based on the analysis of an associated Hamilton--Jacobi--Bellman equation with an Hamiltonian that here, as an innovative fact, depends on both position and momenta.

math.PR↗

arXiv2024

Hamilton--Jacobi equations for Wasserstein controlled gradient flows: existence of viscosity solutions

This work is the third part of a program initiated in arXiv:2111.13258, arXiv:2302.06571 aiming at the development of an intrinsic geometric well-posedness theory for Hamilton-Jacobi equations related to controlled gradient flow problems in metric spaces. In this paper, we finish our analysis in the context of Wasserstein gradient flows with underlying energy functional satisfying McCann's condition. More prescisely, we establish that the value function for a linearly controlled gradient flow problem whose running cost is quadratic in the control variable and just continuous in the state variable yields a viscosity solution to the Hamilton-Jacobi equation in terms of two operators introduced in our former works, acting as rigorous upper and lower bounds for the formal Hamiltonian at hand. The definition of these operators is directly inspired by the Evolutional Variational Inequality formulation of gradient flows (EVI): one of the main innovations of this work is to introduce a controlled version of EVI, which turns out to be crucial in establishing regularity properties, energy and metric bounds along optimzing sequences in the controlled gradient flow problem that defines the candidate solution.

math.AP↗

arXiv2024

Hamilton--Jacobi equations for controlled gradient flows: cylindrical test functions

This work is the second part of a program initiated in arXiv:2111.13258 aiming at the development of an intrinsic geometric well-posedness theory for Hamilton-Jacobi equations related to controlled gradient flow problems in metric spaces. Our main contribution is that of showing that the comparison principle proven therein implies a comparison principle for viscosity solutions relative to smoother Hamiltonians, acting on test functions that are mere cylindrical functions of the underling squared metric distance and whose rigorous definition is achieved from the Evolutional Variational Inequality formulation of gradient flows (EVI). In particular, the new Hamiltonians no longer require to work with test functions containing Tataru's distance. This substantial simplification paves the way for the development of a comprehensive existence theory.

math.AP↗

arXiv2023

Well-posedness of a Hamilton-Jacobi-Bellman equation in the strong coupling regime

We prove comparison principle for viscosity solutions of a Hamilton-Jacobi-Bellman equation in a strong coupling regime considering a stationary and a time-dependent version of the equation. We consider a Hamiltonian that has a representation as the supremum of a difference of two functions: an internal Hamiltonian depending on a control variable and a function interpreted as a cost of applying the controls. Our major innovation lies in the use of a cost function that can be discontinuous, unbounded and depending on momenta, enabling us to address previously unexplored scenarios such as cases arising from the theory of large deviations and homogenisation. For completeness, we also state the existence of viscosity solutions and we verify the assumptions for an example arising from biochemistry.

math.AP↗

arXiv2023

Large deviations for Cox-Ingersoll-Ross processes with state-dependent fast switching

We study the large deviations for Cox-Ingersoll-Ross (CIR) processes with small noise and state-dependent fast switching via associated Hamilton-Jacobi equations. As the separation of time scales, when the noise goes to $0$ and the rate of switching goes to $\infty$, we get a limit equation characterized by the averaging principle. Moreover, we prove the large deviation principle (LDP) with an action-integral form rate function to describe the asymptotic behavior of such systems. The new ingredient is establishing the comparison principle in the singular context. The proof is carried out using the nonlinear semigroup method coming from Feng and Kurtz's book.

math.PR↗

arXiv2021

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 function, both for Glauber dynamics for the Curie-Weiss model and Brownian dynamics in a potential. We extend the variational approach to this problem of time-dependent regularity in order to include Hamiltonian trajectories with a finite lifetime in closed domains with a boundary. This leads to new phenomena, such a recovery of smoothness. We hereby create a new and unifying approach for the study of mean-field Gibbs-non-Gibbs transitions, based on Hamiltonian dynamics and viscosity solutions of Hamilton-Jacobi equations.

math.PR↗

arXiv2020

A large deviation principle for Markovian slow-fast systems

We prove pathwise large deviation principles of slow variables in slow-fast systems in the limit of time-scale separation tending to infinity. In the limit regime we consider, the convergence of the slow variable to its deterministic limit and the convergence of the fast variable to equilibrium are competing at the same scale. The large deviation principle is proven by relating the large deviation problem to solutions of Hamilton-Jacobi-Bellman equations, for which well-posedness was established in the companion paper [arXiv:1912.06579]. We cast the rate functions in action-integral form and interpret the Lagrangians in two ways. First, in terms of a double-optimization problem of the slow variable's velocity and the fast variable's distribution, similar in spirit to what one obtains from the contraction principle. Second, in terms of a principal-eigenvalue problem associated to the slow-fast system. The first representation proves in particular useful in the derivation of averaging principles from the large deviations principles. As main example of our general results, we consider empirical measure-flux pairs coupled to a fast diffusion on a compact manifold. We prove large deviations and use the Lagrangian in double-optimization form to demonstrate the validity of the averaging principle in this system.

math.PR↗

arXiv2020

Comparison Principle for Hamilton-Jacobi-Bellman Equations via a Bootstrapping Procedure

We study the well-posedness of Hamilton-Jacobi-Bellman equations on subsets of $\mathbb{R}^d$ in a context without boundary conditions. The Hamiltonian is given as the supremum over two parts: an internal Hamiltonian depending on an external control variable and a cost functional penalizing the control. The key feature in this paper is that the control function can be unbounded and discontinuous. This way we can treat functionals that appear e.g. in the Donsker-Varadhan theory of large deviations for occupation time measures. To allow for this flexibility, we assume that the internal Hamiltonian and cost functional have controlled growth, and that they satisfy an equi-continuity estimate uniformly over compact sets in the space of controls. In addition to establishing the comparison principle for the Hamilton-Jacobi-Bellman equation, we also prove existence, the viscosity solution being the value function with exponentially discounted running costs. As an application, we verify the conditions on the internal Hamiltonian and cost functional in two examples.

math.AP↗

arXiv2020

Strict continuity of the transition semigroup for the solution of a well-posed martingale problem

In this note we connect the notion of solutions of a martingale problem to the notion of a strongly continuous and locally equi-continuous semigroup on the space of bounded continuous functions equipped with the strict topology. This extends the classical connection of semigroups to Markov processes that was used successfully in the context of compact spaces to the context of Polish spaces. In addition, we consider the context of locally compact spaces and show how the transition semigroup on the space of functions vanishing at infinity can be extended to the space of bounded continuous functions.

math.PR↗

arXiv2020

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 a large deviation principle for the trajectory of the empirical measure and empirical flux pair of such processes. As a second corollary we get same result in the setting where the jump-rates are time-periodic, with period-length that decreases to 0.

math.PR↗

arXiv2020

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 class of associated Hamilton-Jacobi equations. The notable feature for these Hamilton-Jacobi equations is that the Hamiltonian can be discontinuous at the boundary. We prove a well-posedness result for a large class of Hamilton-Jacobi equations corresponding to one-dimensional models, and give partial results for the multi-dimensional setting.

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arXiv2019

The exponential resolvent of a Markov process and large deviations for Markov processes via Hamilton-Jacobi equations

We study the Hamilton-Jacobi equation f - lambda Hf = h, where H f = 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 equation, and which can therefore be interpreted as the resolvent of H. The operator is given in terms of optimization problem where the running cost is a path-space relative entropy. Finally, we use the resolvents to give a new proof of the abstract large deviation result of Feng and Kurtz.

math.PR↗

arXiv2019

Strongly continuous and locally equi-continuous semigroups on locally convex spaces

We consider locally equi-continuous strongly continuous semigroups on locally convex spaces (X,tau). First, we show that if (X,tau) has the property that weak* compact sets of the dual are equi-continuous, then strong continuity of the semigroup is equivalent to weak continuity and local equi-continuity. Second, we consider locally convex spaces (X,tau) that are also equipped with a `suitable' auxiliary norm. We introduce the set N of tau continuous semi-norms that are bounded by the norm. If (X,tau) has the property that N is closed under countable convex combinations, then a number of Banach space results can be generalised in a straightforward way. Importantly, we extend the Hille-Yosida theorem. We apply the results to the study of transition semigroups of Markov processes on complete separable metric spaces.

math.FA↗

arXiv2019

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

We extend the Barles-Perthame procedure of semi-relaxed limits of viscosity solutions of Hamilton-Jacobi equations of the type f - lambda H f = h. The convergence result allows for equations on a `converging sequence of spaces' as well as Hamilton-equations written in terms of two equations in terms of operators H_\dagger and H_\dagger that serve as natural upper and lower bounds for the `true' operator H. In the process, we establish a strong relation between non-linear pseudo-resolvents and viscosity solutions of Hamilton-Jacobi equations. As a consequence we derive a convergence result for non-linear semigroups.

math.FA↗