On the parabolic Martin boundary of the Ornstein-Uhlenbeck operator on Wiener space
Let \(E\) be a separable real Banach space, \(\mu\) be a Gaussian mean zero measure on \((E,{\mathcal B}(E))\) and \((E,H,\mu)\) be an abstract Wiener space. The function \(p(t,\cdot)\) is called \(L\)-parabolic if \(p(t,\cdot)\in L^ 2(E,\mu)\cap D(L)\) and \(Lp(t,\cdot)={d\over dt}p(t,\cdot)\), where \(L\) is a generator of the Ornstein-Uhlenbeck process on \(E\). A positive \(L\)- parabolic function \(p(\cdot,\cdot)\) is called normalized if \(\int_ E p(t,z)\mu(dz)=1\), and let \(\mathcal P\) denote the convex set of all normalized, positive parabolic functions on \(E\). The square-integrable parabolic functions on the space \((E,H,\mu)\) are studied. The author studies the extreme points of the set \(\mathcal P\) and the possibilities to have the integral representation for any \(p\in{\mathcal P}\) in terms of them. The investigation method is similar to the approach developed by Dynkin.
scientific article; zbMATH DE number 89802