How to define (and solve) the diffusion equation with a sticky boundary at the origin? For the diffusion equation $\frac{\partial} {\partial t}  P_t(x)=D \frac{\partial^2} {\partial x^2}  P_t(x)$, a reflecting boundary at the origin for example, means: $\frac{\partial} {\partial x}  P_t(x=0)=0$. 
What is the mathematical way of setting the condition that whenever a particle reaches the origin it stays there forever? 
Note that it does not 'vanish' from the system upon reaching x=0, namely I am not talking about an absorbing boundary. 
Also, how do I solve that differential equation in that case?    
(Thanks for to all helpers!)
 A: I would just take an absorbing boundary condition and then add the absorbed density as a delta function at the sticking point. For convenience, translate the origin so that the sticking point is $x_a>0$ and the particle starts from $x=0$ at $t=0$. The solution then is
$$P(x,t)=f(x,t)-f(2x_a-x,t)+N(t)\delta(x-x_a)$$
$$f(x,t)=(4\pi Dt)^{1/2}e^{-x^2/4Dt}$$
$$N(t)=1-{\rm erf}\,(x_a/\sqrt{4Dt})$$
see for example these lecture notes.
A: Diffusions with partially reflected (including sticky) boundary conditions are discussed in detail in


*

*H. J. Kushner.  Probabilistic methods for finite difference
approximations to degenerate elliptic and parabolic equations with
neumann and dirichlet boundary conditions, J Math Anal Appl 53
(1976), no. 3, 644–668.


Kushner's proof for weak existence/uniqueness of this class of diffusions is based on the submartingale problem formulation developed in 


*

*Stroock, D. W., and Varadhan, S. S. Diffusion processes with
boundary conditions. Communications on Pure and Applied
Mathematics 24.2 (1971): 147-225.


As a byproduct, Kushner also explains how to numerically solve this problem by using the Markov Chain Approximation Method.
A: I agree with kakaz and Carlo Beenakker on stitching together the counted absorbed particles and the absorbing boundary condition solution. Carlo Beenakker quoted the solution for diffusion in 1D starting from a Dirac delta function initial condition. I'd like to add that this problem was also solved in 3D for a uniform initial density in a landmark paper by Smoluchowski in 1917 (http://link.springer.com/10.1007/BF01427232) and was reviewed in English by Chandrasekhar in 1943 (https://link.aps.org/doi/10.1103/RevModPhys.15.1).  The result has been used to provide diffusion-limited reaction rates for chemical reaction rates and annihilation reactions between singlet excitons. Worth a read!
