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?

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!)