$\frac{\partial P}{\partial t} = D \nabla^2 P$,

where D is the diffusion coefficient, which when derived from a random walk is

$D = \frac{\langle \Delta x^2 \rangle}{2dt}$

where $d$ is the dimension of space. The $2$ comes from the fact that walker can jump in any of two directions, per dimension. $\langle \Delta x^2 \rangle$ and $t$ represent the expected distance squared, and the time step in the random walk, respectively. See for example these notes, for derivation.

See also a simple kinetic derivation of diffusion coefficient (in the context of solid state diffusion), see page 7 Also see Alex's notes on kinetic theory.

https://en.wikipedia.org/wiki/Diffusion

Solutions of the diffusion equation

Can show uniqueness.

Solutions to diffusion equation, using Fourier transformation, and using Green functions. Can also derive from Fokker-Planck equation.

Solutions to diffusion equation for free, absorbing, and reflecting boundary conditions.

Solution by Separation of variables

Assume $P=X(x)T(t)$, and get an equation

$\frac{T'}{T}=\frac{X''}{X}$

which implies both sides are equal to the same constant, which can be shown to need to be negative, so: $\frac{T'}{T}=-\alpha^2$ for $T$ and $\frac{X''}{X}=-\alpha^2$

This has solutions

$T(t)=Ce^{-\alpha^2 Dt}$ $X(x)=A\sin{\alpha x} + B\cos{\alpha x}$

and can then apply boundary conditions.