Limit theorems for difference equations in random media with applications to biological systems (original) (raw)

Let (Y; Y) be a measurable space and g : X £ Y ! X be a function which determines the dynamics in a random environment described by a semi-Markov process y(t): Let°b e a small positive parameter and X be a linear space. We consider the dynamical system where states of the system are determined by the following iteration: X°¸(t=°i)+ 1 = X°¸(t=°i) + "g(X°¸(t=°i) ; y¸(t=°i)+ 1); for t 2 R + ; where X°0 = X 0 = x is given,¸(t) is a counting process, i = 1; 2: In this chapter we study: (A) Averaging (i = 1) and di® usion approximation (i = 2) of solutions of the equation as " ! 0 under various assumptions of the data (see [8]);