Semilinear Parabolic Equations, Diffusions, and Superdiffusions

✍ Scribed by E.B. Dynkin

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 361 KB
• Volume: 158
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1998.3298

✦ Synopsis

The semilinear equation u* +L 0 u=u : , where L 0 is a second-order elliptic differential operator without zero-order term and 1<: 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 equation u* +Lu= (u), where Lu=L 0 u+cu (with a bounded coefficient c) and belongs to a convex class which contains ku : with 1<: 2 and positive locally bounded coefficient k. We also cover a substantially wider class of functions which do not correspond to any superdiffusion (for instance, ku : with :>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 equation u* +Lu=& f can be investigated for a class of domains described in probabilistic terms which is substantially larger than the class considered in the literature on PDEs.