On Calabi's strong maximum principle via local semi-Dirichlet forms (Q1930867)

The author defines a notion of \({\mathcal E}\)-subharmonicity in the framework of strong Feller diffusion processes associated to local regular semi-Dirichlet forms with lower bounds. This definition allows a stochastic proof for an extension of E. Calabi's strong maximum principle, even in the absence of a \(C^2\)-differentiable structure. Based on this result, the author shows that his notion of subharmonicity implies a notion of a viscosity subsolution in a stochastic sense. His results apply to singular geometric objects like the Alexandrov space, the limit space under spectral distance of Riemannian manifolds with uniform lower Ricci curvature bound, and so on.