Markov jump process
In the jump process the sample paths $X(t)$ are not smooth but perform jumps.
Definition: Transition rates The transition rate $W(x|x^’,t)$ defines the probabi $W(x|x^’,t)\Delta t$ is the probability density for a jump to $x$ in the infinitesimal time interval $[t,t+\Delta t]$, given that the system is in $x^’$ A total jump rate $\Gamma$ to leave the state $x^’$ can be defined by integrating over $x$: $$\Gamma(x^’, t) = \int \text{d}x W(x|x^’,t)$$
The Generator $\mathcal{A}$ can be defined using a propagator $T$ for infinitesimal time steps $t + \Delta t$ and becomes $$\mathcal{A}(t)\rho(x) = \int \text{d}x^’ \left[W(x|x^’,t)\rho(x^’) - W(x^’|x,t)\rho(x)\right]$$ which defines the differential Chapman-Kolmogorov equation for the propagator and also the density
$$\frac{\partial}{\partial t}T(x,t|x^’,t^’) = \int \text{d}x^{’’}\left[W(x|x^{’’},t)T(x^{’’},t|x^’,t^’)-W(x^{’’}|x,t)T(x,t|x^’,t^’)\right]$$ $$\frac{\partial}{\partial t}\rho(x,t) = \int \text{d}x^{’}\left[W(x|x^{’},t)\rho(x^{’},t)-W(x^{’}|x,t)\rho(x,t)\right]$$
Both equations are called master-equation. Note the difference: $T$ is a conditional probability, $p$ is not, $T$ is neccesarry to define the whole Markov process. The master equations are balance equations (like continuity equation without conservations).
# Homogenous Poisson process
Definition: Poisson process The Poisson process is an integer valued Markov process $X(t) = N(t)$, where in a state $N=n$ only jumps to neighboring states $n\pm 1$
Example: Photon emission of a charged current $\vec{j}(x,t)$. Photons are emitted with a rate $\gamma$
For a general Poisson process with transition rate $\gamma$ the master equation becomes (?) $$\frac{\partial}{\partial t}T(n,t|n^’,t^’) = \gamma T(n-1,t|n’,t’) - \gamma T(n,t|n’,t’)$$
The master equation can be solved with some ansatz. Taking into account that $n \geq n^’$ and $n=n^’$ for $t = t^’$ we get the propagator $$T(n,t|n^’,t^’)=\frac{[\gamma(t-t^’)]^{n-n^’}}{(n-n’)\exp(-\gamma (t-t^’)).$$ This propagator only depends on the differences in time and $n$.
The probability distribution of $n$ jumps happening over the time $t$ is thererfore the known Poisson distribution $$P(n,t)=\frac{(\gamma t)^n}{n!}\exp(-\gamma t), n\geq 0,$$ with expectation value und variance beeing $\gamma t$.
# Non-homogenous Poisson process
The rate $\gamma$ can depend on the time $\gamma = \gamma(t)$.
Again calculating the propagator gives $$T(n,t|n^’,t^’)=\frac{[\mu(t-t^’)]^{n-n^’}}{(n-n’)\exp(-\mu(t-t^’)),$$ with $$\mu(t,t^’)=\int^{t}_{t^’}\text{d}s \gamma(s),$$ and therefore $$P(n,t)=\frac{(\mu(t,0)^n}{n!}\exp(-\mu(t,0), n\geq 0,$$ with $\text{E}(N(t)) = \text{Var}(N(t)) = \mu(t,0)$.