A Shorokhod problem with singular drift and its application to the origin of universes
type: Item
instance of: scholarly article
title: A Shorokhod problem with singular drift and its application to the origin of universes
zbMATH Open document ID: 0814.60077
author: Thomas Domenig
author: Masao Nagasawa
published in: Proceedings of the Japan Academy. Series A
publication date: 1994-12-18T00:00:00Z
review text: From the authors: Let \(R(t)\) be strictly increasing and continuous in \(t \geq 0\) with \(R(0) = 0\). In a space time domain \[ D = \biggl\{ (t,x) : t > 0, \;x \in \bigl[ - R(t), R(t) \bigr] \biggr\}, \tag \(*\) \] we consider a singular diffusion and its formal adjoint: \[ {\partial u \over \partial t} + {1 \over 2} \sigma^ 2 {\partial^ 2u \over \partial x^ 2} + {x \over t} {\partial u \over \partial x} = 0, \quad - {\partial \mu \over \partial t} + {1 \over 2} \sigma^ 2 {\partial^ 2 \mu \over \partial x^ 2} - {\partial \over \partial x} \left( {x \over t} \mu \right) = 0 \] with the reflecting boundary condition. \((*)\) determines a transition probability \(Q(s,x; t,dy)\), \(s,t \in [a,b]\), \(0 < a < b < \infty\). Since \(\{Q(s,x; t,dy)\), \(s \in (0, \varepsilon]\}\) is tight because of \((*)\), we can choose \(\xi (s) \downarrow 0\) so that \(Q^ \xi (0,0; t,dy) = \lim_{s \downarrow 0} Q(s, \xi (s); t,dy)\) exists, but the limit depends on \(\xi\) and is not uniquely determined in general. We discuss this problem and its implication to the origin of universes in terms of a Skorokhod problem with singular drift \(x/t\).
reviewed by: Malempati M. Rao
Mathematics Subject Classification ID: 60J60
Mathematics Subject Classification ID: 60J65
Mathematics Subject Classification ID: 60H15
zbMATH DE Number: 701457
zbMATH Keywords: origin of universes
zbMATH Keywords: singular diffusion
zbMATH Keywords: reflecting boundary condition
zbMATH Keywords: Skorokhod problem
MaRDI profile type: Publication
full work available at URL: pjaa.70.88
OpenAlex ID: W1979626151
cites work: Q3139889
cites work: Q3477773
cites work: Q4693741
DOI: 10.3792/PJAA.70.88
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description: scientific article; zbMATH DE number 701457