Sharp spectral asymptotics for non-reversible metastable diffusion processes
| hal.structure.identifier | Institut Denis Poisson [IDP] | |
| dc.contributor.author | LE PEUTREC, Dorian | |
| hal.structure.identifier | Institut de Mathématiques de Bordeaux [IMB] | |
| dc.contributor.author | MICHEL, Laurent | |
| dc.date.accessioned | 2024-04-04T02:49:31Z | |
| dc.date.available | 2024-04-04T02:49:31Z | |
| dc.date.issued | 2020 | |
| dc.identifier.issn | 2690-0998 | |
| dc.identifier.uri | https://oskar-bordeaux.fr/handle/20.500.12278/191828 | |
| dc.description.abstractEn | 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 $L=-h\Delta+U_h\cdot\nabla$ under the assumptions that $U_h=U_{0}+h\nu$, where the vector fields $U_{0}:\mathbb R^{d}\to \mathbb R^{d}$ and $\nu:\mathbb R^{d}\to \mathbb R^{d}$ 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 $\epsilon>0$ such that in the limit $h\to 0$, $L$ admits exactly $n_0$ eigenvalues in the strip $\{0\leq \operatorname{Re}(z)< \epsilon\}$, 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. | |
| dc.description.sponsorship | Analyse Quantitative de Processus Metastables - ANR-19-CE40-0010 | |
| dc.language.iso | en | |
| dc.publisher | MSP | |
| dc.subject.en | Non-reversible overdamped Langevin dynamics | |
| dc.subject.en | Metastability | |
| dc.subject.en | Non-reversible diffusion processes | |
| dc.subject.en | Spectral theory | |
| dc.subject.en | Semiclassical analysis | |
| dc.subject.en | Eyring-Kramers formulas | |
| dc.title.en | Sharp spectral asymptotics for non-reversible metastable diffusion processes | |
| dc.type | Article de revue | |
| dc.identifier.doi | 10.2140/pmp.2020.1.3 | |
| dc.subject.hal | Mathématiques [math]/Théorie spectrale [math.SP] | |
| dc.subject.hal | Mathématiques [math]/Equations aux dérivées partielles [math.AP] | |
| dc.subject.hal | Mathématiques [math]/Physique mathématique [math-ph] | |
| bordeaux.journal | Probability and Mathematical Physics | |
| bordeaux.page | 3-53 | |
| bordeaux.volume | 1 | |
| bordeaux.hal.laboratories | Institut de Mathématiques de Bordeaux (IMB) - UMR 5251 | * |
| bordeaux.issue | 1 | |
| bordeaux.institution | Université de Bordeaux | |
| bordeaux.institution | Bordeaux INP | |
| bordeaux.institution | CNRS | |
| bordeaux.peerReviewed | oui | |
| hal.identifier | hal-02189630 | |
| hal.version | 1 | |
| hal.popular | non | |
| hal.audience | Internationale | |
| hal.origin.link | https://hal.archives-ouvertes.fr//hal-02189630v1 | |
| bordeaux.COinS | ctx_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.jtitle=Probability%20and%20Mathematical%20Physics&rft.date=2020&rft.volume=1&rft.issue=1&rft.spage=3-53&rft.epage=3-53&rft.eissn=2690-0998&rft.issn=2690-0998&rft.au=LE%20PEUTREC,%20Dorian&MICHEL,%20Laurent&rft.genre=article |
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