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August 2020 On stiff problems via Dirichlet forms
Liping Li, Wenjie Sun
Ann. Inst. H. Poincaré Probab. Statist. 56(3): 2051-2080 (August 2020). DOI: 10.1214/19-AIHP1028


The stiff problem is concerned with a thermal conduction model with a singular barrier of zero volume. In this paper, we shall build the phase transitions for the stiff problems in one-dimensional space. It turns out that every phase transition definitely depends on the total thermal resistance of the barrier, and the three phases correspond to the so-called impermeable pattern, semi-permeable pattern and permeable pattern of thermal conduction respectively. For each pattern, the related boundary condition of the flux at the barrier is also derived. Mathematically, we shall introduce and explore the so-called snapping out Markov process, which is the probabilistic counterpart of semi-permeable pattern in the stiff problem.

Le problème raide (the « Stiff problem ») concerne un modèle de diffusion de la chaleur avec une barrière singulière de volume zéro. Dans ce papier, nous établissons le diagramme de phase pour le problème raide en dimension 1 : il y a trois phases (la phase imperméable, la phase demi-perméable et la phase perméable) et chaque phase dépend de la résistance thermique totale de la barrière. De plus, pour chaque phase, nous identifions la condition au bord pour le flux sur la barrière. Du point de vue mathématique, nous introduisons et étudions le processus de Markov à transition brusque (the snapping out Markov process), qui donne une interprétation probabiliste de la phase demi-perméable dans le problème raide.


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Liping Li. Wenjie Sun. "On stiff problems via Dirichlet forms." Ann. Inst. H. Poincaré Probab. Statist. 56 (3) 2051 - 2080, August 2020.


Received: 26 May 2018; Revised: 29 September 2019; Accepted: 1 October 2019; Published: August 2020
First available in Project Euclid: 26 June 2020

MathSciNet: MR4116717
Digital Object Identifier: 10.1214/19-AIHP1028

Primary: 31C25 , 60J25 , 60J45 , 60J50

Keywords: Dirichlet forms , Mosco convergences , Phase transitions , Snapping out Markov processes , Stiff problems

Rights: Copyright © 2020 Institut Henri Poincaré


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Vol.56 • No. 3 • August 2020
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