A diffusion equation

Abstract

For a » 1 and s » 1 we have obtained accurate estimates of the equilibrium joint distribution when $\kappa = {{\left( {s - a} \right)} \mathord{\left/ {\vphantom {{\left( {s - a} \right)} {\sqrt s }}} \right. \kern-\nulldelimiterspace} {\sqrt s }}$ is either large and positive or large and negative. In the former case, N_i(t) could be treated like a continuous random variable but the distribution was sensitive to the integer values of N_s(t). In the latter case N_s(t) could be treated as a continuous random variable but the distribution of N_i(t) decayed at a geometric rate (like (s/a)^j’). From the arguments of section 3 one can also understand how these distributions evolve in time. If κ is of order 1, however, N_i(t) and N_s(t) should both be of order $\sqrt s$ and behave like continuous random variables except possibly on or near boundaries where N_i(t) and/or N_s(t) = 0.