On Transition Densities of Singularly Perturbed Diffusions with Fast and Slow Components

We derive asymptotic properties of transition densities for singularly perturbed diffusion processes with fast and slow components. Our study focuses on the Kolmogorov–Fokker–Planck equations. The model can be viewed as a diffusion process having two time scales and is motivated by a wide variety of applications involving singularly perturbed Markov processes in manufacturing systems, homogenization, reliability analysis, queueing networks, statistical physics, population biology, financial economics, and many other related fields. By virtue of the methods of matched singular perturbation, asymptotic expansion is constructed for the transition density. The expansion includes both regular part and boundary layer corrections. Detailed justification of the asymptotic expansion is given, and error bounds are also provided.

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