An Exact Solution for a Class of Stochastic Partial Differential Equations

A brief and personal history of stochastic partial differential equations

“…The product σ(u) ξ is not always well-defined, since in many cases of interest ξ is a generalised function and u is not expected to be smooth; in this case one writes the equation in an integral form and uses Itô integration to give a sense to the stochastic term. The idea of associating PDEs and randomness was already present in the physics literature in the '50s and '60s, see for example [87,64,19,42]. In the mathematical literature, several authors extended Itô's theory of stochastic differential equations (SDE) to a Hilbert space setting, see for example Daleckiȋ [27] and Gross [45].…”