# Probability Density Function of Random Variable

I have the jump Markov process $$(N_z)_{z\geq0}$$ which behaves like a symmetric random walk on teh set of positive integers. For $$z$$ large enough, $$N_z=0$$ where $$0$$ is an absorbing state. I now define a random variable:

$$T_\infty = 2\int_{0}^{\infty}N_z dz$$

I want to write the probability density function $$P^{\infty}_p$$ of the random variable $$T_\infty$$, with the initial condition $$N_0 = p$$. I then want to show it satisfies the differential equation

$$\frac{dP^{\infty}_p}{d\tau} = \frac{p}{2}(P^{\infty}_{p+1}-2P^{\infty}_p)+P^{\infty}_{p-1}),$$ where $$p\geq1.$$

How do I tackle this?