Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Construction and analysis of a sticky reflected distorted Brownian motion

Torben Fattler, Martin Grothaus, and Robert Voßhall

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Abstract

We give a Dirichlet form approach for the construction of a distorted Brownian motion in $E:=[0,\infty)^{n}$, $n\in\mathbb{N}$, where the behavior on the boundary is determined by the competing effects of reflection from and pinning at the boundary (sticky boundary behavior). In providing a Skorokhod decomposition of the constructed process we are able to justify that the stochastic process is solving the underlying stochastic differential equation weakly for quasi every starting point with respect to the associated Dirichlet form. That the boundary behavior of the constructed process indeed is sticky, we obtain by proving ergodicity of the constructed process. Therefore, we are able to show that the occupation time on specified parts of the boundary is positive. In particular, our considerations enable us to construct a dynamical wetting model (also known as Ginzburg–Landau dynamics) on a bounded set $D_{N}\subset\mathbb{Z}^{d}$ under mild assumptions on the underlying pair interaction potential in all dimensions $d\in\mathbb{N}$. In dimension $d=2$ this model describes the motion of an interface resulting from wetting of a solid surface by a fluid.

Résumé

Nous construisons un mouvement brownien tordu dans $E:=[0,\infty)^{n}$, $n\in\mathbb{N}$, en utilisant des méthodes de la théorie des formes de Dirichlet alors que le comportement à la frontière est déterminé par les effets concurrents de la réflexion de la frontière et l’ancrage à la frontière (comportement adhésif sur la frontière de $E$). En fournissant une décomposition de Skorokhod du processus construit nous pouvons justifier que le processus stochastique est une solution faible de l’équation différentielle stochastique fondamentale pour quasi tous les points de départ par rapport à la forme de Dirichlet associée. En démontrant l’ergodicité du processus construit, nous obtenons que le comportement sur la frontière du processus est en effet adhésif. En conséquence, il est possible de démontrer que le séjour sur des parties fixées de la frontière de $E$ est positif. En particulier, nos considérations nous permettent de construire un modèle dynamique d’humectage (ausssi connu comme dynamique de Ginzburg–Landau) sur un ensemble borné $D_{N}\subset\mathbb{Z}^{d}$, $d\in\mathbb{N}$. Le potentiel qui détermine l’interaction des variables adjacentes est soumis à des conditions peu restrictives en toute dimension $d\in\mathbb{N}$. En dimension $d=2$, ce modèle décrit le mouvement d’une interface résultant de l’humectage d’une surface solide par un fluide.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 52, Number 2 (2016), 735-762.

Dates
Received: 18 September 2013
Revised: 11 September 2014
Accepted: 30 September 2014
First available in Project Euclid: 4 May 2016

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1462367892

Digital Object Identifier
doi:10.1214/14-AIHP650

Mathematical Reviews number (MathSciNet)
MR3498008

Zentralblatt MATH identifier
1342.60166

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60J50: Boundary theory 60J55: Local time and additive functionals 82C41: Dynamics of random walks, random surfaces, lattice animals, etc. [See also 60G50]

Keywords
Interacting sticky reflected distorted Brownian motion Skorokhod decomposition Wentzell boundary condition Interface models

Citation

Fattler, Torben; Grothaus, Martin; Voßhall, Robert. Construction and analysis of a sticky reflected distorted Brownian motion. Ann. Inst. H. Poincaré Probab. Statist. 52 (2016), no. 2, 735--762. doi:10.1214/14-AIHP650. https://projecteuclid.org/euclid.aihp/1462367892


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