Stochastic Differential Equations and Related Inverse Problems

Abstract

AbstractAs 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.KeywordsBrownian MotionHide LayerStochastic Differential EquationWiener ProcessQuadratic VariationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.