This work is the third part of a program initiated in [12,11] 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 Evolution 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.

This work is the second part of a program initiated in [13] 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 underlying 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.

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 contributes to the literature in three ways: (1) We cast the Crandall–Ishii Lemma into a test function framework to tackle a wide class of second-order integro-differential operators in the spirit of the classical doubling of variables method. (2) We provide a unified approach to estimate the difference of Hamiltonians by adapting the probabilistic notion of couplings to an analytic setting. (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.

We consider a class of slow–fast processes on a connected complete Riemannian manifold M. The limiting dynamics as the scale separation goes to ∞ 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 Hamilton–Jacobi–Bellman (HJB) equation techniques. Our main innovation is solving the comparison principle for viscosity solutions for the HJB equation on M and the construction of a variational viscosity solution for the non-smooth Hamiltonian, which lies at the heart of deriving the action integral representation for the rate function.

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 homogenises over the periodic potential as well as over the internal degree of freedom. Around the homogenised 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.

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 of the regularizing properties of gradient flows in evolutional variational inequality (EVI) formulation, that we exploit for constructing rigorous upper and lower bounds for the formal Hamiltonian at hand and, in combination with the use of the Tataru's distance, for establishing the key estimates needed to bound the difference of the Hamiltonians in the proof of the comparison principle. Our abstract results apply to a large class of examples only partially covered by the existing theory, including gradient flows on Hilbert spaces and the Wasserstein space equipped with a displacement convex energy functional E satisfying McCann's condition.

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 terms of operators H_† and H_‡ 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.

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-periodic rates in which the period-length converges to 0.

We study the well-posedness of Hamilton–Jacobi–Bellman equations on subsets of 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.

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.

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 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.

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.

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 behavior of reversible stochastic processes to the gradient-flow structure of their deterministic limit. Natural examples of application include particle systems with inertia.

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 in terms of an 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 (2006).

The author regrets a mistake made in Kraaij [2]. We summarize the results which remain valid and those whose validity is now unclear.
An overview of the status of the main results: Let X be a separable metric space. On X we consider the space of bounded continuous functions [Formula presented] equipped with the strict topology, cf. Sentilles [3]. In addition, let [Formula presented] be the space of τ-additive Borel measures on X and let σ be the weak topology on [Formula presented] induced by [Formula presented]. In Kraaij [2], four additional topologies were considered on [Formula presented]: • σ ^lf, the finest locally convex topology on [Formula presented] that coincides with σ on all β-equicontinuous sets in [Formula presented],• σ ^f, the finest topology on [Formula presented] that coincides with σ on all β-equicontinuous sets in [Formula presented],• kσ the finest topology on [Formula presented] that coincides with σ on all weakly compact sets in [Formula presented],• β ^∘ the polar topology on [Formula presented] generated using all pre-compact sets in [Formula presented], cf. Köthe [1].The main result of Kraaij [2] is Theorem 1.7 that states that σ ^lf = σ ^f = kσ = β ^∘. The following result remains true Proposition 1.1 σ ^f = kσ and σ ^lf = β ^∘. As a consequence of the missing identification σ ^f = σ ^lf, it is unclear whether [Formula presented] is infra-Pták by using Proposition 1.2. This in turn leads to the failure of establishing Corollaries 1.10, 1.11 and 1.12. Proposition 1.6 and Lemma's 1.8 and 1.9 are established using results in the literature and remain valid as it is. The mistake and an overview of its consequences in the proof sections: The proof that σ ^f = σ ^lf was based on the observation that as σ ^lf⊆σ ^f it suffices to verify that σ ^f is locally convex. This was carried out in two steps. Step 1: σ ^f was explicitly identified as a quotient topology T and it was shown that kσ = T = σ ^f. This part remains valid. Step 2: The explicit characterization T was then used to establish that T is locally convex. This part contains an error in the proof of Lemma 2.7. As a consequence, is unclear whether Lemma 2.8 and Proposition 2.6 remain true. The error: The proof that addition is a continuous map for T is mistaken, the proof that scalar multiplication is continuous remains valid. The exact mistake in Lemma 2.7 is the claim that H⊆U. Let [Formula presented] be the map defined by [Formula presented]. Let A, B be σ ₊ open subsets in [Formula presented] for and let C be σ open in M _τ. Finally let [Formula presented]. The sets H and U were defined as [Formula presented] The issue is that ⊕ adds the set C interpreted as the diagonal [Formula presented] to the set A×B, whereas the construction to obtain H adds the much larger product space C×C to [Formula presented]. As a consequence the claim H⊆U remains unproven.           

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 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.

Much of the structure of macroscopic evolution equations for relaxation to equilibrium can be derived from symmetries in the dynamical fluctuations around the most typical trajectory. For example, detailed balance as expressed in terms of the Lagrangian for the path-space action leads to gradient zero-cost flow. We expose a new such fluctuation symmetry that implies GENERIC, an extension of gradient flow where a Hamiltonian part is added to the dissipative term in such a way as to retain the free energy as Lyapunov function.