Invariant measures and symmetry property of Lévy type operators
Infinitesimal invariant probability measures of elliptic integro-differential operators on \(\mathbb{R}^d\) of type \[ Lf= a_1\Delta f+ a_2(-\Delta)^{\alpha/2}f+ \beta\cdot\nabla f,\qquad f\in{\mathcal S}(\mathbb{R}^d), \] with \(a_1,a_2\in [0,\infty)\), \(a_1+ a_2> 0\), \(\beta: \mathbb{R}^d\to \mathbb{R}^d\) are studied, i.e. measures \(\mu\) satisfying the equation \(L^*\mu= 0\). Under some conditions existence of regular infinitesimal invariant probability measures is proved. It is also shown that for such a measure \(\mu(L,{\mathcal S}(\mathbb{R}^d))\) can only be symmetric on \(L^2(\mathbb{R}^d,\mu)\) if \(a_2= 0\).
infinitesimal invariant probability measures
scientific article; zbMATH DE number 1546779