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.@en

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 in [JaKu17]. 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 (containing precisely two spin values). 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. @en

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 in [JaKu17]. 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 (containing precisely two spin values). 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. @en

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.            @en

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.@en

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.@en

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. @en

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.@en

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 unique invariant measure. In the case of Gaussian single-spin distribution, we analyze the dynamics of moderate fluctuations for the magnetization. We obtain a path-space moderate deviation principle via a general analytic approach based on convergence of non-linear generators and uniqueness of viscosity solutions for associated Hamilton-Jacobi equations. Our result shows that, under a peculiar moderate space-time scaling and without tuning external parameters, the typical behavior of the magnetization is critical.@en

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 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.@en