A probabilistic approach to the equation \(Lu=-u^2\) (Q1968737)

Let \(L\) be a second-order uniformly elliptic differential operator in \(\mathbb{R}^d\) with bounded smooth coefficients such that \(L[1]= 0\). In a recent work, the author introduced some subclass \({\mathcal U}_1(D)\) of positive solutions of the differential equation \(L[u]= -u^2\) which he related by a certain integral equation in a 1-1 manner to some convex subclass \({\mathcal H}_1(D)\) of the positive \(L\)-harmonic functions. In the present paper he provides a probabilistic characterization of the classes \({\mathcal U}_1(D)\) and \({\mathcal H}_1(D)\). This characterization is based on the notion of a superdiffusion. Similar results are obtained in the parabolic setting.