Singularly Perturbed Multidimensional Switching Diffusions with Fast and Slow Switchings

This work is concerned with singularly perturbed switching diffusions with fast and slow components. The diffusion component of the joint process is multidimensional and the diffusive motion evolves an order of magnitude slower than that of the jumps or the switchings. The jump component may be decomposed into several groups. There is one group consisting of transient states, and all other groups involve recurrent states. Within each group, there are rapid switchings. Among different groups, the switchings occur relatively infrequently. Under simple conditions, an asymptotic expansion of the probability density is developed; error bounds and justification of the expansion are provided.