A mini course on stochastic partial differential equations

Abstract

Partial differential equations constitute one of the most effective and important tools in the mathematical modelling and the literature devoted to their theory and applications is enormous. It was the wave equation,

$$\begin{array}{*{20}{c}} {\frac{{{{\partial }^{2}}u}}{{\partial {{t}^{2}}}}(t,\xi ) = \frac{{{{\partial }^{2}}u}}{{\partial {{\xi }^{2}}}}(t,\xi ),} & {t \geqslant 0,\xi \in {{R}^{1}},} \\ \end{array}$$

which was first formulated and studied. It appeared around 1740 in the works of J.R. d’Alembert, D. Bernolli and L. Euler. The theory of the heat equation was

$$\begin{array}{*{20}{c}} {\frac{{\partial u}}{{\partial t}}(t,\xi ) = \frac{{{{\partial }^{2}}u}}{{\partial {{\xi }^{2}}}}(t,\xi ),} & {t \geqslant 0,\xi \in (0,1),} \\ \end{array}$$

initiated later by J. Fourier in his book “Théorie analytique de la chaleur” published in 1822.