We consider a Feller branching diffusion process X with drift c having 0 as a slowly reflecting (sticky) boundary point with a stickiness parameter . We show that (i) the process X can be characterised as a unique weak solution to the SDE system
where and are given and fixed, B is a standard Brownian motion, and is a diffusion local time process of X at 0, and (ii) the transition density function of X can be expressed in the closed form by means of a convolution integral involving a new special function and a modified Bessel function of the second kind. The new special function embodies the stickiness of X entirely and reduces to the Mittag-Leffler function when . We determine a (sticky) boundary condition at zero that characterises the transition density function of X as a unique solution to the Kolmogorov forward/backward equation of X. Letting (absorption) and (instantaneous reflection) the closed-form expression for the transition density function of X reduces to the ones found by Feller  and Molchanov  respectively. The results derived for sticky Feller diffusions translate over to yield closed-form expressions for the transition density functions of (a) sticky Cox-Ingersoll-Ross processes and (b) sticky reflecting Vasicek processes that can be used to model slowly reflecting interest rates.
Dedicated to the memory of William Feller (1906–1970)
"Sticky Feller diffusions." Electron. J. Probab. 28 1 - 28, 2023. https://doi.org/10.1214/23-EJP909