【Mathathon 1】The Wiener Criterion at ∞ for the Elliptic and Parabolic PDEs and its Measure-Theoretical, Topological and Probabilistic Consequences.

Speaker: Prof. Ugur G. Abdulla, Analysys and Partial Differential Equations Unit, OIST

Title: The Wiener Criterion at ∞ for the Elliptic and Parabolic PDEs and its Measure-Theoretical, Topological and Probabilistic Consequences.

Abstract: 
Wiener’s celebrated result on the boundary regularity of harmonic functions is one of the most beautiful and delicate results of 20th-century mathematics. It has shaped the boundary regularity theory for elliptic and parabolic PDEs and has become a central result in the development of potential theory at the intersection of functional analysis, PDE, probability and measure theories. This lecture presents my solution of the Kolmogorov’s Problem (1928) expressed in the Wiener-type criterion for the regularity of ∞ for the second-order elliptic and parabolic PDEs in arbitrary open sets. The new concept of regularity or irregularity of ∞ for the elliptic (or parabolic) PDEs is defined according to whether or not the harmonic (or caloric) measure of ∞ is null or positive. Wiener-type criterion precisely characterizes the regularity of the point at ∞ for the second-order elliptic and parabolic PDEs and broadly extends the Wiener test’s role in Analysis. It arises as a global characterization of uniqueness in boundary value problems for arbitrary unbounded open sets. From a topological point of view, the Wiener test at ∞ arises as a thinness criterion at ∞ in fine topology. In a probabilistic context, the Wiener test at ∞ characterizes asymptotic laws for the characteristic Markov processes whose generator is the given differential operator. The counterpart of the new Wiener test at a finite boundary point leads to uniqueness in the class of functions growing at a certain rate near the boundary point; the criterion for the characterization of singularities of elliptic and parabolic PDEs; thinness criterion near Martin boundary point at minimal-fine topology; asymptotic laws for the conditional Brownian motion; The talk will end with the description of some outstanding open problems and perspectives of the development of the potential theory of nonlinear elliptic and parabolic PDEs.