This book is written by a well-known specialist in the theory of Markov processes and partial differential equations and is devoted to applications of probability theory to the theory of positive solutions of the equation Lu = (u), where L is an elliptic differential operator of second order and is a differentiable positive function. The equation under consideration contains as a particular case the equation u = u, whose positive solutions were studied by M. Marcus and L. Veron with purely analytical methods. The author has applied both analytic and probabilistic approaches. The Choquet capacities are one of the principal tools for the study of the equation u = u. This probabilistic tool is closely connected with the use of super diffusion theory. The book will be useful for anybody interested in applications of probabilistic methods to mathematical analysis.