arXiv subjects

Boris Nectoux

Publications and source records attributed to Boris Nectoux.

22 records · Page 2Linked to original sources

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↗

arXiv2018

Sharp estimate of the mean exit time of a bounded domain in the zero white noise limit

We prove a sharp asymptotic formula for the mean exit time from a bounded domain $D\subset \mathbb R^d$ for the overdamped Langevin dynamics $$d X_t = -\nabla f(X_t) d t + \sqrt{2\ve} \ d B_t$$ when $\ve \to 0$ and in the case when $D$ contains a unique non degenerate minimum of $f$ and $\pa_{\mbf n}f>0$ on $\pa D$. This formula was actually first derived in~\cite{matkowsky-schuss-77} using formal computations and we thus provide, in the reversible case, the first proof of it. As a direct consequence, we obtain when $\ve \to 0$, a sharp asymptotic estimate of the smallest eigenvalue of the operator $$L_{\ve}=-\ve Δ+\nabla f\cdot \nabla$$ associated with Dirichlet boundary conditions on $\pa D$. The approach does not require $f|_{\partial D}$ to be a Morse function. The proof is based on results from~\cite{Day2,Day4} and a formula for the mean exit time from $D$ introduced in~\cite{BEGK, BGK}.

math.AP↗

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↗