Uniqueness of invariant measures and essential \(m\)-dissipativity of diffusion operators on \(L^1\) (Q2759733)

It is proved that there exists at most one probability measure \(\mu\) on \(\mathbb{R}^d\), so that \(L^*\mu= 0\), where \(L= a^{ij}\partial_i\partial_j+ b^i\partial_i\), provided \((L,C^\infty_0(\mathbb{R}^d))\) is essential \(m\)-dissipativity on \(L^1(\mathbb{R}^d,\nu)\) for at least one \(\nu\), so that \(L^*\nu= 0\). Here it is assumed that \((a^{ij})\) is nondegenerate, \(a^{ij}\in H^{(p,1)}_{\text{loc}}\), and \(b^i\in L^p_{\text{loc}}\). We also present a whole class of examples (even for \(a^{ij}= \delta^{ij}\)), where \(L^*\mu= 0\) has more than one solution. Furthermore, recent related results are reviewed.NEWLINENEWLINEFor the entire collection see [Zbl 0968.00044].