Local WKB construction for boundary Witten Laplacians

40 pagesInternational audienceWKB $p$-forms are constructed as approximate solutions to boundary value problems associated with semi-classical Witten Laplacians. Naturally distorted Neumann or Dirichlet boundary conditions are considered

Small eigenvalues of the Neumann realization of the semiclassical Witten Laplacian

Troisième version, 69 pagesInternational audienceThis article follows the previous works \cite{HKN} by Helffer-Klein-Nier and \cite{HelNi1} by Helffer-Nier about the metastability in reversible diffusion processes via a Witten complex approach. Again, exponentially small eigenvalues of some self-adjoint realization of $\Delta_{f,h}^{(0)}=-h^{2}\Delta +\left|\nabla f(x)\right|^{2}-h\Delta f(x)\;,$ are considered as the small parameter $h>0$ goes to $0$. The function $f$ is assumed to be a Morse function on some bounded domain $\Omega$ with boundary $\partial\Omega$. Neumann type boundary conditions are considered. With these boundary conditions, some simplifications possible in the Dirichlet problem studied in \cite{HelNi1} are no more possible. A finer treatment of the three geometries involved in the boundary problem (boundary, metric, Morse function) is carried out

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

Let f : R d → R be a smooth function and (Xt) t≥0 be the stochastic process solution to the overdamped Langevin dynamics dXt = −−f (Xt)dt + √ h dBt. Let Ω ⊂ 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 → 0 + , the asymptotic repartition of the quasi-stationary distribution of (Xt) t≥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 → 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 → 0 + of the first exit point distribution from Ω of (Xt) t≥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≥0 be the stochastic process solution to the overdamped Langevin dynamics in R d : dX t = −−f (X t)dt + √ h dB t , (1) where f : R d → R is the potential (chosen C ∞ in all this work), h > 0 is the temperature and (B t) t≥0 is a standard d-dimensional Brownian motion. Let Ω be a C ∞ bounded open and connected subset of R d and introduce τ Ω = inf{t ≥ 0 | X t / ∈ 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 ∼ µ h , it holds for any time t > 0 and any Borel set A ⊂ Ω, P(X t ∈ A | t < τ Ω) = µ h (A)

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

Precise Arrhenius law for p-forms: The Witten Laplacian and Morse-Barannikov complex

Accurate asymptotic expressions are given for the exponentially small eigenvalues of Witten Laplacians acting on p-forms. The key ingredient, which replaces explicit formulas for global quasimodes in the case p = 0, is Barannikov's presentation of Morse theory

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 $\Omega \subset \mathbb R^d$ be the basin of attraction of a local minimum of $f: \mathbb R^d \to \mathbb R$. Up to a small perturbation of $\Omega$ to make it smooth, we prove that the exit rates of $(X_t)_{t\ge 0}$ from $\Omega$ through each of the saddle points of $f$ on $\partial \Omega$ 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

Small singular values of an extracted matrix of Witten complex

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On Witten Laplacians and Brascamp–Lieb’s Inequality on Manifolds with Boundary

International audienceIn 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