A Shorokhod problem with singular drift and its application to the origin of universes
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 \(x/t\).
scientific article; zbMATH DE number 701457
scientific article