Differential equations in which the Poisson process plays a role

Kac observes that this is the restatement, in the language of continuous stochastic processes, of a result of Goldstein [2], who, starting with a Poisson-type random walk, asymptotically obtained solutions of (1). The proof in [ l ] goes via direct computation, which may be done also in the case of the Laplacian in several space dimensions. Tha t some sort of similar result must hold in even greater generality, and with a more elegant proof, was suggested by Professor Kac, to whom the author is indeed grateful. The purpose of this note is to present such a result.