Full metastable asymptotic of the Fisher information

We establish an expansion by Gamma-convergence of the Fisher information relative to the reference measure exp(-beta V), where V is a generic multiwell potential and beta goes to infinity. The expansion reveals a hierarchy of multiple scales reflecting the metastable behavior of the underlying overdamped Langevin dynamics: distinct scales emerge and become relevant depending on whether one considers probability measures concentrated on local minima of V, probability measures concentrated on critical points of V, or generic probability measures on R^d. We thus fully describe the asymptotic behavior of minima of the Fisher information over regular sets of probabilities. The analysis mostly relies on spectral properties of diffusion operators and the related semiclassical Witten Laplacian and covers also the case of a compact smooth manifold as underlying space.Comment: 24 pages. Typos correcte

Spectral Analysis of Discrete Metastable Diffusions

We consider a discrete Schr\"odinger operator $H_\varepsilon= -\varepsilon^2\Delta_\varepsilon + V_\varepsilon$ on $\ell^2(\varepsilon \mathbb Z^d)$, where $\varepsilon>0$ is a small parameter and the potential $V_\varepsilon$ is defined in terms of a multiwell energy landscape $f$ on $\mathbb R^d$. This operator can be seen as a discrete analog of the semiclassical Witten Laplacian of $\mathbb R^d$. It is unitarily equivalent to the generator of a diffusion on $\varepsilon \mathbb Z^d$, satisfying the detailed balance condition with respect to the Boltzmann weight $\exp{(-f/\varepsilon)}$. These type of diffusions exhibit metastable behaviour and arise in the context of disordered mean field models in Statistical Mechanics. We analyze the bottom of the spectrum of $H_\varepsilon$ in the semiclassical regime $\varepsilon\ll1$ and show that there is a one-to-one correspondence between exponentially small eigenvalues and local minima of $f$. Then we analyze in more detail the bistable case and compute the precise asymptotic splitting between the two exponentially small eigenvalues. Through this purely spectral-theoretical analysis of the discrete Witten Laplacian we recover in a self-contained way the Eyring-Kramers formula for the metastable tunneling time of the underlying stochastic process

An Eyring-Kramers law for the stochastic Allen-Cahn equation in dimension two

We study spectral Galerkin approximations of an Allen--Cahn equation over the two-dimensional torus perturbed by weak space-time white noise of strength $\sqrt{\varepsilon}$. We introduce a Wick renormalisation of the equation in order to have a system that is well-defined as the regularisation is removed. We show sharp upper and lower bounds on the transition times from a neighbourhood of the stable configuration $-1$ to the stable configuration $1$ in the asymptotic regime $\varepsilon \to 0$. These estimates are uniform in the discretisation parameter $N$, suggesting an Eyring-Kramers formula for the limiting renormalised stochastic PDE. The effect of the "infinite renormalisation" is to modify the prefactor and to replace the ratio of determinants in the finite-dimensional Eyring-Kramers law by a renormalised Carleman-Fredholm determinant.Comment: 28 pages, 1 Figure. Revised version with expanded discussio

Jump Markov models and transition state theory: the Quasi-Stationary Distribution approach

We are interested in the connection between a metastable continuous state space Markov process (satisfying e.g. the Langevin or overdamped Langevin equation) and a jump Markov process in a discrete state space. More precisely, we use the notion of quasi-stationary distribution within a metastable state for the continuous state space Markov process to parametrize the exit event from the state. This approach is useful to analyze and justify methods which use the jump Markov process underlying a metastable dynamics as a support to efficiently sample the state-to-state dynamics (accelerated dynamics techniques). Moreover, it is possible by this approach to quantify the error on the exit event when the parametrization of the jump Markov model is based on the Eyring-Kramers formula. This therefore provides a mathematical framework to justify the use of transition state theory and the Eyring-Kramers formula to build kinetic Monte Carlo or Markov state models.Comment: 14 page

Best Translation Practices of Medical Language Terms in the Works of Galdós: the Example of Fortunata y Jacinta

In this paper, through the study of some terms present in the novel Fortunata y Jacinta by Galdós that refer to the medical and pharmaceutical field, we want to carry out a reflection on some ‘best practices’ that the translator can use in his rewriting work, considered as the result of a cognitive operation, a complex intrigue of acquisition, categorization, reflective observation, elaboration, experimentation and creation of emotional-motivational dynamics which has, as its ultimate product, the recreation of the metatext. These 'best practices' fall to all intents and purposes in the topics to be treated within a teaching of translation with a neurolinguistic slant.In quest’articolo, attraverso lo studio di alcuni termini presenti nel romanzo Fortunata y Jacinta di Galdós che si riferiscono all’ambito medico e farmaceutico, si desidera portare avanti una riflessione su alcune “buone prassi” che il traduttore può impiegare nel suo lavoro di riscrittura, considerate come il frutto di un’operazione cognitiva, complesso intrigo di acquisizione, categorizzazione, osservazione riflessiva, elaborazione, sperimentazione e di creazione di dinamiche emotivo-motivazionali che hanno come prodotto ultimo la ricreazione del metatesto. Tali “buone prassi” rientrano a tutti gli effetti negli argomenti da trattare all’interno di una didattica della traduzione dal taglio neurolinguistico

L'"OSTINATO RIGORE" DEL TRADUTTORE

The reflection on the role of the translator, on his way of approaching a prototext, on the responsibility of constructing a metatext can only be described in the Leonardian terms of an "Ostinato Rigore", or in terms of method and will. The rigor, therefore, the method can be seen as the sum of the skills, abilities and competences that allow the translator to manage the translation process. It is in the actual translation in which the "rigor" becomes "obstinate", that is, the method merges with the desire to seek the most adequate rendering in the target text and it is in this cognitive operation that psychophysiological components come into play such as memory and emotional-motivational dynamics

Simboli religiosi e libertà religiosa. Profili costituzionali

The essay analyzes some aspects of the freedom of religion enshrined in the Italian Constitution in relation to the principle of secularism as supreme principle of the Italian Republic, according to the definition of the italian Constitutional Court; and the extension of religious symbols and, in particular, the crucifix. First of all, the essay analyzes the question of «Italian» secularism, verifying the disparities and similarities among the various declinations of this principle, reconstructing the debate in the constituent assembly and the main jurisprudential approaches of the Italian Constitutional Court in this matter. Secondly, the essay analyzes the issue of religious symbols in public spaces and, in particular, a leading case concerning the exhibition of the crucifix in the classrooms, both at the jurisprudence level (Administrative Judge, Constitutional Court, EDU Court), and at doctrinal one. The essay, then, notes the cd. Bavarian solution, adopted by the Government of the homonymous German region, in search of a reconciliation between the religious freedom of each person and the quantitative factor per single religion. There is also an excursus on the main case law of the European Court of Human Rights about secularity, religious freedom and religious symbols