Stochastic Space-Time Models and Limit Theorems: An Introduction

Abstract

Stochastic space-time models describe phenomena which change with time, are inhomogeneous in space and depend on chance. They arise in various fields of physics, chemistry, biology and engineering. Both analogy to the theory of spatially homogeneous stochastic systems and other reasoning suggest that a class of stochastic space-time phenomena could be modelled as solutions of SPDE’s which can be formally written as a (deterministic) partial differential equation (PDE) perturbed by a “suitable” noise term

$$ \frac{\partial }{{\partial t}}X\left( {t,r,\omega } \right) = P\left( {t,r,X,{\partial _i}X} \right) + \xi \left( {t,r,X,{\partial _i}X,\omega } \right) $$

(1)

(initial condition, boundary condition - if necessary).