Nonlinear PDEs and measure-valued branching type processes

We deal with the probabilistic approach to a nonlinear operator Lambda of the form Lambda u = Delta u + Sigma(infinity)(k=1) qk(uk), in connection with the works of M. Nagasawa, N. Ikeda, S. Watanabe, and M.L. Silverstein on the discrete branching processes. Instead of the Laplace operator we may consider the generator of a right (Markov) process, called base process, with a general (not necessarily locally compact) state space. It turns out that solutions of the nonlinear equation Lambda u = 0 are produced by the harmonic functions with respect to the (linear) generator of a discrete branching type process. The consideration of the general state space allows to take as base process a measure-valued superprocess (in the sense of E.B. Dynkin). The probabilistic counterpart is a Markov process which is a combination between a continuous branching process (e.g., associated with a nonlinear operator of the form Delta u - u(alpha), 1 < alpha <= 2) and a discrete branching type one, on a space of configurations of finite measures. Our approach uses probabilistic and analytic potential theoretical tools, like the potential kernel of a continuous additive functional and the subordination operators. (C) 2010 Elsevier Inc. All rights reserved.