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?


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