Asymptotic Properties of Parabolic Systems for Null-Recurrent Switching Diffusions

This work is concerned with the asymptotic behavior of systems of parabolic equations arising from null-recurrent switching diffusions,which are diffusion processes modulated by continuous-time Markov chains. A sufficient condition for null recurrence is presented.Moreover,convergence rate of the solutions of systems of homogeneous parabolic equations under suitable conditions is established.Then a case study on verifying one of the conditions proposed is provided with the use of a two-state Markov chain.To verify the condition,boundary value problems (BVPs) for parabolic systems are treated,which are not the usual two-point BVP type.An extra condition in the interior is needed resulting in jump discontinuity of the derivative of the corresponding solution.