It⊚ Diffusions, Modified Capacity, and Harmonic Measure. Applications to Schrödinger Operators

We observe that some special It⊚ diffusions are related to scattering properties of a Schrödinger operator on $\mathbb R^d,\ d\geq 2$. We introduce Feynman–Kac type formulae for these stochastic processes, which lead us to the results on the preservation of the absolutely continuous spectrum of the Schrödinger operator. To better understand the analytic properties of the processes, we construct and study a special version of the potential theory. The modified capacity and harmonic measure play an important role in these considerations. Various applications to Schrödinger operators are also given. For example, we relate the presence of the absolutely continuous spectrum to the geometric properties of the support of the potential.