Semilinear Parabolic Equations, Diffusions, and Superdiffusions

The semilinear equationu+L0u=uα, whereL0is a second-order elliptic differential operator without zero-order term and 1<α⩽2, has been studied by the author in [4] and [5] by using superdiffusions. In the present paper, we apply superdiffusions to a more general equationu+Lu=ψ(u), whereLu=L0u+cu(with a bounded coefficientc) andψbelongs to a convex class which containskuαwith 1<α⩽2 and positive locally bounded coefficientk. We also cover a substantially wider class of functionsψwhich do not correspond to any superdiffusion (for instance,kuαwithα>1). Related problems are treated with the help of diffusion processes. This approach is useful even in the linear theory. For instance, the first boundary value problem for equationu+Lu=−fcan be investigated for a class of domains described in probabilistic terms which is substantially larger than the class considered in the literature on PDEs.