Connection between the Burgers equation with an elastic forcing term and a stochastic process

Abstract

We present a complete analytical resolution of the one dimensional Burgers equation with the elastic forcing term $-{\kappa }^{2}x+f(t)$, $\kappa \in \mathbb{R}$. Two methods existing for the case $\kappa =0$ are adapted and generalized using variable and function transformations, valid for all values of space an time. The emergence of a Fokker-Planck equation in the method allows to connect a fluid model, depicted by the Burgers equation, with an Ornstein-Uhlenbeck process.