A limit theorem for branching one-dimensional periodic diffusion processes (Q794060)

One dimensional branching diffusion processes whose non-branching part is ''a periodic diffusion process'' are considered to obtain their asymptotic behavior of the particle numbers in a bounded domain. The emphasized points are: (1) the corresponding semigroup \(T_ t\) is non-compact, (2) the principal eigenvalue for the process is positive, (3) the spectral functions are unbounded. The authors employed the eigenfunction expansion theory and \(L^ p\)-martingale method for \(1<p\leq 2\). Some spectral properties of periodic diffusion operators including Hill's one are also studied.