Random evolution equations in hydrology

Abstract

A number of partial differential equations governing the dynamic behavior of different components of the hydrologic cycle in a region can be treated as abstract evolution equations in appropriate Hilbert spaces. When the source terms, the boundary conditions, the initial conditions, and/or the equation parameters are prescribed as stochastic processes, the resulting random evolution equations may be solved in terms of the input stochastic processes if the semigroup associated with the partial differential operator is known. This places in the hands of the modeler a very flexible tool to solve a large variety of problems. Several explicit examples of the methodology are illustrated. Finally, applications of the Itǒ's lemma in Hilbert spaces to the numerical solution of stochastic partial differential equations in hydrology are presented.