Stochastic Differential Equations and Related Inverse Problems

As we have discussed in Chap. 1, the deterministic mathematical formulation of solute transport through a porous medium introduces the dispersivity, which is a measure of the distance a solute tracer would travel when the mean velocity is normalized to be one. One would expect such a measure to be a mechanical property of the porous medium under consideration, but the evidence are there to show that dispersivity is dependent on the scale of the experiment for a given porous medium. One of the challenges in modelling the phenomena is to discard the Fickian assumptions, through which dispersivity is defined, and develop a mathematical description containing the fluctuations associated with the mean velocity of a physical ensemble of solute particles. To this end, we require a sophisticated mathematical framework, and the theory of stochastic processes and differential equations is a natural mathematical setting. In this chapter we review some essential concepts in stochastic processes and stochastic differential equations in order to understand the stochastic calculus in a more applied context.