I am considering a one-dimensional process represented by a diffusion equation.
$$\partial_tP=D \partial_x^2P$$
The probability distribution $P(x,t)$ obeys that $P(x,0)=\delta(x-l)$ for a given value of $l>0$. We consider this distribution in the interval $[0,d]$.
I want to calculate the probability of getting to the point $x=d$ subject to the condition of not having been at 0 at any previous time. How can I calculate this probability? Without the condition I thought that I can express the condition of getting to $x=d$ as:
$$P(d)=\int_0^\infty P(d,t) dt$$
but I don't know how to incorporate the supplementary condition about 0.