Asymptotic Behavior of Parabolic Equations Arising from One-Dimensional Null-Recurrent Diffusions

This work is concerned with the asymptotic behavior of homogeneous and nonhomogeneous parabolic equations arising from one-dimensional null-recurrent diffusion processes. First, we review the concepts of regularity, recurrence, and positive recurrence of Markov processes and recall the connections of these concepts with properties of solutions of the corresponding differential equations. Next, we examine the rate of convergence of the solutions of both homogeneous and nonhomogeneous parabolic equations when the initial function and the forcing function are integrable with respect to the invariant measure. Weaker and verifiable conditions compared with the existing work in the literature are obtained. Then the corresponding problems when the initial and forcing functions are not integrable with respect to the invariant measure are dealt with. Convergence under suitable scaling for the solutions of the parabolic equations is proved, and the explicit limit is obtained.