arXiv subjects

Giacomo Di Gesù

Publications and source records attributed to Giacomo Di Gesù.

12 recordsLinked to original sources

arXiv2026

$L^2$ convergence rates for singular stochastic wave dynamics

We establish quantitative hypocoercivity for singular stochastic damped wave equations arising in stochastic quantization. Building on the finite-dimensional argument of Cao, Lu, and Wang [Arch. Rat. Mech. Anal. 2023], we derive a dimension-free criterion that replaces uniform semiconvexity by a log-Sobolev inequality together with exponential lower-tail bounds for the bottom of the Hessian spectrum. We verify this criterion for the two-dimensional $Φ^4_2$ and sine-Gordon models and for the three-dimensional $Φ^4_3$ model, through estimates on the lowest eigenvalue of associated singular random Schrödinger operators. Combined with the continuum limit established in a companion paper, this yields exponential $L^2$ convergence to equilibrium for the $Φ^4_2$ dynamics and for sine-Gordon with $β^2<2π$. For $Φ^4_2$, we verify that in a weak-coupling, small-mass regime, optimizing the damping yields a kinetic relaxation rate of at least order $m$. This yields a genuine acceleration over the order-$m^2$ relaxation rate of the overdamped dynamics. The lattice estimates are uniform in the discretization for sine-Gordon throughout $β^2<4π$, and for $Φ^4_3$ at fixed volume, including regimes beyond those for which the continuum dynamics is vpresently constructed.

math-ph↗

arXiv2026

Eyring-Kramers asymptotics for infinite-dimensional stochastic gradient systems

We study small-noise asymptotics for a class of reversible stochastic evolution equations in infinite dimensions. The dynamics are of the form \[ dX_t=-A\nabla F(X_t)\,dt+\sqrt{2β^{-1}A}\,dW_t, \] where $F$ is a regular multi-well potential, $A$ is a selfadjoint mobility operator, $W$ is a cylindrical Brownian motion and $β\gg 1$ is the inverse noise strength. The invariant measure is a Gibbs perturbation of a Gaussian reference measure, and the resulting framework covers, in particular, the stochastic Allen-Cahn and stochastic Cahn-Hilliard equations on bounded intervals. In the double-well case, we derive a sharp asymptotic formula for the first nonzero eigenvalue of the generator. This gives an infinite-dimensional Eyring-Kramers law for the spectral gap, with exponential rate determined by the communication height and leading prefactor determined by the local quadratic behavior at the relevant minima and saddle points. Our approach provides a general strategy for lifting finite-dimensional Eyring-Kramers analysis to infinite-dimensional stochastic gradient systems.

math.PR↗

arXiv2024

Asymptotics of the $ϕ^4_1$ measure in the sharp interface limit

We consider the $ϕ^4_1$ measure in an interval of length $\ell$, defined by a symmetric double-well potential $W$ and inverse temperature $β$. Our results concern its asymptotic behavior in the joint limit $β, \ell \to \infty$, both in the subcritical regime $\ell \ll \rme^{βC_W}$ and in the supercritical regime $\ell \gg \rme^{βC_W}$, where $C_W$ denotes the surface tension. In the former case, in which the measure concentrates on the pure phases, we prove the corresponding large deviation principle. The associated rate function is the Modica-Mortola functional modified to take into account the entropy of the locations of the interfaces. Further, we provide the sharp asymptotics of the probability of having a given number of transitions between the two pure phases. In the supercritical regime, the measure does not longer concentrate and we show that the interfaces are asymptotically distributed according to a Poisson point process.

math.PR↗

arXiv2021

Spectral Analysis of Discrete Metastable Diffusions

We consider a discrete Schrödinger operator $ H_\varepsilon= -\varepsilon^2Δ_\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.

math-ph↗

arXiv2019

Sharp Tunneling Estimates for a Double-Well Model in Infinite Dimension

We consider the stochastic quantization of a quartic double-well energy functional in the semiclassical regime and derive optimal asymptotics for the exponentially small splitting of the ground state energy. Our result provides an infinite-dimensional version of some sharp tunneling estimates known in finite dimensions for semiclassical Witten Laplacians in degree zero. From a stochastic point of view it proves that the $L^2$ spectral gap of the stochastic one-dimensional Allen-Cahn equation in finite volume satisifies a Kramers-type formula in the limit of vanishing noise. We work with finite-dimensional lattice approximations and establish semiclassical estimates which are uniform in the dimension. Our key estimate shows that the constant separating the two exponentially small eigenvalues from the rest of the spectrum can be taken independently of the dimension.

math.PR↗

arXiv2019

The exit from a metastable state: concentration of the exit point distribution on the low energy saddle points

We consider the first exit point distribution from a bounded domain $Ω$ of the stochastic process $(X_t)_{t\ge 0}$ solution to the overdamped Langevin dynamics $$d X_t = -\nabla f(X_t) d t + \sqrt{h} \ d B_t$$ starting from the quasi-stationary distribution in $Ω$. In the small temperature regime ($h\to 0$) and under rather general assumptions on $f$ (in particular, $f$ may have several critical points in $Ω$), it is proven that the support of the distribution of the first exit point concentrates on some points realizing the minimum of $f$ on $\partial Ω$. The proof relies on tools to study tunnelling effects in semi-classical analysis. Extensions of the results to more general initial distributions than the quasi-stationary distribution are also presented.

math.AP↗

arXiv2019

Sharp asymptotics of the first exit point density

We consider the exit event from a metastable state for the overdamped Langevin dynamics $dX_t = -\nabla f(X_t) dt + \sqrt{h} dB_t$. Using tools from semiclassical analysis, we prove that, starting from the quasi stationary distribution within the state, the exit event can be modeled using a jump Markov process parametrized with the Eyring-Kramers formula, in the small temperature regime $h \to 0$. We provide in particular sharp asymptotic estimates on the exit distribution which demonstrate the importance of the prefactors in the Eyring-Kramers formula. Numerical experiments indicate that the geometric assumptions we need to perform our analysis are likely to be necessary. These results also hold starting from deterministic initial conditions within the well which are sufficiently low in energy. From a modelling viewpoint, this gives a rigorous justification of the transition state theory and the Eyring-Kramers formula, which are used to relate the overdamped Langevin dynamics (a continuous state space Markov dynamics) to kinetic Monte Carlo or Markov state models (discrete state space Markov dynamics). From a theoretical viewpoint, our analysis paves a new route to study the exit event from a metastable state for a stochastic process.

math.AP↗

arXiv2018

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.

math.PR↗

arXiv2017

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.

math.PR↗

arXiv2016

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.

math.PR↗

arXiv2015

Small noise spectral gap asymptotics for a large system of nonlinear diffusions

We study the $L^2$ spectral gap of a large system of strongly coupled diffusions on unbounded state space and subject to a double-well potential. This system can be seen as a spatially discrete approximation of the stochastic Allen-Cahn equation on the one-dimensional torus. We prove upper and lower bounds for the leading term of the spectral gap in the small temperature regime with uniform control in the system size. The upper bound is given by an Eyring-Kramers-type formula. The lower bound is proven to hold also for the logarithmic Sobolev constant. We establish a sufficient condition for the asymptotic optimality of the upper bound and show that this condition is fulfilled under suitable assumptions on the growth of the system size. Our results can be reformulated in terms of a semiclassical Witten Laplacian in large dimension.

math.SP↗

arXiv2010

Small noise asymptotic of the Gallavotti-Cohen functional for diffusion processes

We consider, for a diffusion process in R^n, the Gallavotti-Cohen functional, defined as the empirical power dissipated in a time interval by the non-conservative part of the drift. We prove a large deviation principle in the limit in which the noise vanishes and the time interval diverges. The corresponding rate functional, which satisfies the fluctuation theorem, is expressed in terms of a variational problem on the classical Freidlin-Wentzell functional.

math.PR↗