Searcharxiv⌕ Search

arXiv subjects

Dorian Le Peutrec

Publications and source records attributed to Dorian Le Peutrec.

18 recordsLinked to original sources
arXiv2025

Eyring-Kramers formula for the mean exit time of non-Gibbsian elliptic processes: the non characteristic boundary case

In this work, we derive a new sharp asymptotic equivalent in the small temperature regime $h\to 0$ for the mean exit time from a bounded domain for the non-reversible process $dX\_t=b(X\_t)dt + \sqrt h \, dB\_t$ under a generic orthogonal decomposition of $b$ and when the boundary of $Ω$ is assumed to be \textit{non characteristic}. The main contribution of this work lies in the fact that we do not assume that the process $(X\_t,t\ge 0)$ is \textit{Gibbsian}. In this case, a new correction term characterizing the \textit{non-Gibbsianness} of the process appears in the equivalent of the mean exit time. The proof is mainly based on tools from spectral and semi-classical analysis.

math.AP↗
arXiv2023

Exit time and principal eigenvalue of non-reversible elliptic diffusions

In this work, we analyse the metastability of non-reversible diffusion processes $$dX_t=\boldsymbol{b}(X_t)dt+\sqrt h\,dB_t$$ on a bounded domain $Ω$ when $\mathbf{b}$ admits the decomposition $\mathbf{b}=-(\nabla f+\mathbf{\ell})$ and $\nabla f \cdot \mathbf{\ell}=0$. In this setting, we first show that, when $h\to 0$, the principal eigenvalue of the generator of $(X_t)_{t\ge 0}$ with Dirichlet boundary conditions on the boundary $\partialΩ$ of $Ω$ is exponentially close to the inverse of the mean exit time from $Ω$, uniformly in the initial conditions $X_0=x$ within the compacts of $Ω$. The asymptotic behavior of the law of the exit time in this limit is also obtained. The main novelty of these first results follows from the consideration of non-reversible elliptic diffusions whose associated dynamical systems $\dot X=\mathbf{b}(X)$ admit equilibrium points on $\partialΩ$. In a second time, when in addition $÷\mathbf{\ell} =0$, we derive a new sharp asymptotic equivalent in the limit $h\to 0$ of the principal eigenvalue of the generator of the process and of its mean exit time from $Ω$. Our proofs combine tools from large deviations theory and from semiclassical analysis, and truly relies on the notion of quasi-stationary distribution.

math.PR↗
arXiv2022

Power of the Crowd

Consider an Galton Watson tree of height $m$: each leaf has one of $k$ opinions or not. In other words, for $i \in \{1, . . . , k\}$, $x$ at generation $m$ thinks $i$ with probability $p_i$ and nothing with probability $p_0$. Moreover the opinions are independently distributed for each leaf. Opinions spread along the tree based on a specific rule: the majority wins! In this paper, we study the asymptotic behavior of the distribution of the opinion of the root when $m \to +\infty$.

math.PR↗
arXiv2022

Eyring-Kramers exit rates for the overdamped Langevin dynamics: the case with saddle points on the boundary

Let $(X_t)_{t\ge 0}$ be the stochastic process solution to the overdamped Langevin dynamics $$dX_t=-\nabla f(X_t) \, dt +\sqrt h \, dB_t$$ and let $Ω\subset \mathbb R^d $ be the basin of attraction of a local minimum of $f: \mathbb R^d \to \mathbb R$. Up to a small perturbation of $Ω$ to make it smooth, we prove that the exit rates of $(X_t)_{t\ge 0}$ from $Ω$ through each of the saddle points of $f$ on $\partial Ω$ can be parametrized by the celebrated Eyring-Kramers laws, in the limit $h \to 0$. This result provides firm mathematical grounds to jump Markov models which are used to model the evolution of molecular systems, as well as to some numerical methods which use these underlying jump Markov models to efficiently sample metastable trajectories of the overdamped Langevin dynamics.

math.PR↗
arXiv2022

Eyring-Kramers type formulas for some piecewise deterministic Markov processes

In this work, we give sharp asymptotic equivalents in the small temperature regime of the smallest eigenvalues of the generator of some piecewise deterministic Markov processes (including the ZigZag process and the Bouncy Particle Sampler process) with refreshment rate $α$ on the one-dimensional torus T. These asymptotic equivalents are usually called Eyring-Kramers type formulas in the literature. The case when the refreshment rate $α$ vanishes on T is also considered.

math-ph↗
arXiv2022

Eyring-Kramers law for Fokker-Planck type differential operators

We consider Fokker-Planck type differential operators associated with general Langevin processes admitting a Gibbs stationary distribution. Under assumptions insuring suitable resolvent estimates, we prove Eyring-Kramers formulas for the bottom of the spectrum of these operators in the low temperature regime. Our approach is based on the construction of sharp Gaussian quasimodes which avoids supersymmetry or PT-symmetry assumptions.

math.AP↗
arXiv2020

The exit from a metastable state: concentration of the exit point distribution on the low energy saddle points, part 2

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 deterministic initial conditions in $Ω$, under rather general assumptions on $f$ (for instance, $f$ may have several critical points in $Ω$). This work is a continuation of the previous paper \cite{DLLN-saddle1} where the exit point distribution from $Ω$ is studied when $X_0$ is initially distributed according to the quasi-stationary distribution of $(X_t)_{t\ge 0}$ in $Ω$. The proofs are based on analytical results on the dependency of the exit point distribution on the initial condition, large deviation techniques and results on the genericity of Morse functions.

math.AP↗
arXiv2020

Sharp spectral asymptotics for non-reversible metastable diffusion processes

Let $U_h:\mathbb R^{d}\to \mathbb R^{d}$ be a smooth vector field and consider the associated overdamped Langevin equation $$dX_t=-U_h(X_t)\,dt+\sqrt{2h}\,dB_t$$ in the low temperature regime $h\rightarrow 0$. In this work, we study the spectrum of the associated diffusion $L=-hΔ+U_h\cdot\nabla$ under the assumptions that $U_h=U_{0}+hν$, where the vector fields $U_{0}:\mathbb R^{d}\to \mathbb R^{d}$ and $ν:\mathbb R^{d}\to \mathbb R^{d}$ are independent of $h\in(0,1]$, and that the dynamics admits $e^{-\frac Vh}$ as an invariant measure for some smooth function $V:\mathbb{R}^d\rightarrow\mathbb{R}$. Assuming additionally that $V$ is a Morse function admitting $n_0$ local minima, we prove that there exists $ε>0$ such that in the limit $h\to 0$, $L$ admits exactly $n_0$ eigenvalues in the strip $\{0\leq \operatorname{Re}(z)< ε\}$, which have moreover exponentially small moduli. Under a generic assumption on the potential barriers of the Morse function $V$, we also prove that the asymptotic behaviors of these small eigenvalues are given by Eyring-Kramers type formulas.

math.SP↗
arXiv2020

Small eigenvalues of the Witten Laplacian with Dirichlet boundary conditions: the case with critical points on the boundary

In this work, we give sharp asymptotic equivalents in the limit $h\to 0$ of the small eigenvalues of the Witten Laplacian, that is the operator associated with the quadratic form $$ ψ\in H^1_0(Ω)\mapsto h^2 \int_Ω\big \vert \nabla \big (e^{\frac 1hf} ψ\big )\big \vert^2\, e^{-\frac 2hf},$$where $\overlineΩ=Ω\cup \partial Ω$ is an oriented $C^\infty$ compact and connected Riemannian manifold with non empty boundary $\partial Ω$ and $f: \overline Ω\to \mathbb R$ is a $C^\infty$ Morse function. The function $f$ is allowed to admit critical points on $ \partial Ω$, which is the main novelty of this work in comparison with the existing literature.

math.SP↗
arXiv2020

Bar codes of persistent cohomology and Arrhenius law for p-forms

This article shows that counting or computing the small eigenvalues of the Witten Laplacian in the semi-classical limit can be done without assuming that the potential is a Morse function as the authors did in [LNV]. In connection with persistent cohomology, we prove that the rescaled logarithms of these small eigenvalues are asymptotically determined by the lengths of the bar code of the function f. In particular, this proves that these quantities are stable in the C 0 topology on the space of functions. Additionally, our analysis provides a general method for computing the subexponential corrections in a large number of cases.

math.AP↗
arXiv2019

Repartition of the quasi-stationary distribution and first exit point density for a double-well potential

Let f : R d $\rightarrow$ R be a smooth function and (Xt) t$\ge$0 be the stochastic process solution to the overdamped Langevin dynamics dXt = ----f (Xt)dt + $\sqrt$ h dBt. Let $Ω$ $\subset$ R d be a smooth bounded domain and assume that f | $Ω$ is a double-well potential with degenerate barriers. In this work, we study in the small temperature regime, i.e. when h $\rightarrow$ 0 + , the asymptotic repartition of the quasi-stationary distribution of (Xt) t$\ge$0 in $Ω$ within the two wells of f | $Ω$. We show that this distribution generically concentrates in precisely one well of f | $Ω$ when h $\rightarrow$ 0 + but can nevertheless concentrate in both wells when f | $Ω$ admits sufficient symmetries. This phenomenon corresponds to the so-called tunneling effect in semiclassical analysis. We also investigate in this setting the asymptotic behaviour when h $\rightarrow$ 0 + of the first exit point distribution from $Ω$ of (Xt) t$\ge$0 when X0 is distributed according to the quasi-stationary distribution. 1 Setting and results 1.1 Quasi-stationary distribution and purpose of this work Let (X t) t$\ge$0 be the stochastic process solution to the overdamped Langevin dynamics in R d : dX t = ----f (X t)dt + $\sqrt$ h dB t , (1) where f : R d $\rightarrow$ R is the potential (chosen C $\infty$ in all this work), h > 0 is the temperature and (B t) t$\ge$0 is a standard d-dimensional Brownian motion. Let $Ω$ be a C $\infty$ bounded open and connected subset of R d and introduce $τ$ $Ω$ = inf{t $\ge$ 0 | X t / $\in$ $Ω$} the first exit time from $Ω$. A quasi-stationary distribution for the process (1) on $Ω$ is a probability measure $μ$ h on $Ω$ such that, when X 0 $\sim$ $μ$ h , it holds for any time t > 0 and any Borel set A $\subset$ $Ω$, P(X t $\in$ A | t < $τ$ $Ω$) = $μ$ h (A).

math.AP↗
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

Exit event from a metastable state and Eyring-Kramers law for the overdamped Langevin dynamics

In molecular dynamics, several algorithms have been designed over the past few years to accelerate the exit event from a metastable region of the configuration space. Some of them are based on the fact that the exit event from a metastable region is well approximated by a Markov jump process. In this work, we present recent results on the exit event from a metastable region for the overdamped Langevin dynamics obtained in [17, 18, 49]. These results aim in particular at justifying the use of a Markov jump process parametrized by the Eyring-Kramers law to model the exit event from a metastable region.

math.AP↗
arXiv2017

On Witten Laplacians and Brascamp-Lieb's inequality on manifolds with boundary

In this paper, we derive from the supersymmetry of the Witten Laplacian Brascamp-Lieb's type inequalities for general differential forms on compact Riemannian manifolds with boundary. In addition to the supersymmetry, our results essentially follow from suitable decompositions of the quadratic forms associated with the Neumann and Dirichlet self-adjoint realizations of the Witten Laplacian. They moreover imply the usual Brascamp-Lieb's inequality and its generalization to compact Riemannian manifolds without boundary.

math.SP↗
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↗