Monotonicity of the reflected Bessel transition density on the diagonal

For α ∈ R , let p R ( t , x , x ) denote the diagonal of the transition density of the α -Bessel process in ( 0 , 1 ] , killed at 0 and reflected at 1. As a function of x , if either α ≥ 3 or α = 1 , then for t > 0 , the diagonal is nondecreasing. This monotonicity property fails if 1 ≠ α < 3 .