Abstract
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.
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This research was supported in part by the National Science Foundation under DMS-0603287, in part by the National Security Agency, MSPF-068-029, and in part by the National Natural Science Foundation of China under No.60574069.
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Khasminskii, R.Z., Zhu, C. & Yin, G. Asymptotic Properties of Parabolic Systems for Null-Recurrent Switching Diffusions. Acta Mathematicae Applicatae Sinica, English Series 23, 177–194 (2007). https://doi.org/10.1007/s10255-007-0362-7
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DOI: https://doi.org/10.1007/s10255-007-0362-7