On Calabi’s Strong Maximum Principle via Local Semi-Dirichlet Forms

We give a stochastic proof of an extension of E. Calabi??s strong maximum principle under some geometric conditions in the framework of strong Feller diffusion processes associated to local regular semi-Dirichlet forms with lower bounds. As a corollary, our notion of subharmonicity implies a notion of viscosity subsolution in a stochastic sense. We can apply our result to singular geometric object like Alexandrov space, limit space under spectral distance of Riemannian manifolds with uniform lower Ricci curvature bound and so on.