Setup:
A tool I have used as a tool to learn about stochastic processes has been to solve deterministic problems with stochastic machinery. For example, for a Brownian process with drift $$ \dot{x} = V + \xi(t)$$ we have a Fokker-Planck equation $$ \partial_t P(x,t|x_0) = (-V \partial_x + D \partial_x^2 )P(x,t|x_0).$$ If we turn off the diffusion and make the problem deterministic, we get a simple advection equation: $$ (\partial_t + V \partial_x)P = 0$$ which has solution $P(x,t|x_0) = \delta(x-x_0-V t).$ So we have solved for the distribution of this deterministic process.
Problem statement:
I would like to continue this way of thinking and get the (deterministic) first passage time distribution. Therfore I want to solve $(\partial_t + V \partial_x)P(x,t|x_0)=0$ with the initial condition $P(x,0|x_0) = \delta(x-x_0)$, where $x_0<0$, and the absorbing boundary condition $P(0,t|x_0)=0$. I then want to compute the survival probability $S(t|x_0) = \int_0^t P(0,t|x_0)dt$, and hence the first passage time distribution from $x=x_0$ to $x=0$, via $f(t|x_0) = -\partial_t S(t|x_0)$. I anticipate the result $$ f(t) = \delta(t- x_0/V)\Theta(-x_0),$$ but I am not sure how to obtain this.
Attempt:
The advection equation $(\partial_t + V \partial_x)P(x,t|x_0)=0$ has characteristic curves $x_0 = x- Vt$. Therefore I construct a solution $$ P(x,t|x_0) = F(x - Vt).$$ This needs to meet the boundary condition $$ F(-Vt) = 0$$ and the initial condition $$ F(x) = \delta(x-x_0).$$ Here's the struggle. I don't know how to solve this pair of equations. I guess together they imply $\delta(-Vt - x_0) = 0,$ which is true if $-x_0 < Vt)$, but I'm really not sure here. Any advice?