Probability density function for a Brownian motion in confined geometries

Question

In the classical Brownian motion, the probability density function of finding a given particle in $x$ at time $t$, given that it is at $y$ can be expressed as:

$$p(t,x,y)=\frac{1}{2\pi \sigma^{2} t} e^{- \frac{|x-y|^{2}}{2\sigma^{2}t}}.$$

This expression assume an infinite two-dimensional space.

I was wondering whether there exists the corresponding probability density function for the case of a confined geometry, let's say a disk of radius R or other simple confined geometries such as a square or rectangle. The problem would be 'What is the $p(t,x,y)$
if a particle is constrained within a circle with reflecting boundaries?'

I have just discovered a couple of papers in which they have solved this problem and provided the probability density function, in terms of polar coordinates: Eq. 27 in https://www.frontiersin.org/articles/10.3389/fphy.2020.583202/full and Eq. 9 in 'A note to on confined diffusion' by Bickel, 2007.

As written in Mortensen et al., 2021 Eq. 27 "For the case of Brownian motion confined to a 2D disc, the solution to the diffusion Fokker–Planck equation for the conditional probability density for the particle’s position ${\bf r}$ at time $t$, given it was ${\bf r_0}$ at time $t=0$, is, with closed boundary conditions at $r$=$a$,"

$$p({\bf r},t|{\bf r_0},t_0)=\frac{1}{\pi a^{2}} + \frac{1}{\pi a^{2}} \sum_{l=-\infty}^\infty cos[l(\phi-\phi_0)] \sum_{m=1}^\infty \frac{\alpha_{lm}^2}{\alpha_{lm}^2 -l^2} e^{-\alpha_{lm}^2\frac{t}{\tau}} \frac{J_l(\alpha_{lm}\frac{r}{a})J_l(\alpha_{lm}\frac{r_0}{a})}{J_l(\alpha_{lm})^2}.$$

"where $\tau=a^2/D$ and $\alpha_{lm}$ > 0 is the $m$th positive root, $J′(\alpha_{lm})=0$, of the derivative of the Bessel function of the first kind of order $l$, $J_l$. The zeros are arranged in ascending order of magnitude: $0<\alpha_{l1}<\alpha_{l2}<....$"

Note that the particle's position is expressed in polar coordinates as ${\bf r}={\bf r}(r,\phi)$ with $0≤r<a$ and $0≤\phi<2\pi$.

It is however unclear to me what some of the parameters of the equation mean. I know that $a$ is the radius of the disc, $\phi$ and $\phi_0$ are defined in the polar coordinates, $D$ is the diffusion coefficient, but I do not understand how to compute $\alpha_{lm}$. Here it says that it is the "the $m$th positive root, $J′(\alpha_{lm})=0$, of the derivative of the Bessel function of the first kind of order $l$, but I do not understand what this means.