In probability theory, the telegraph process is a memoryless continuous-time stochastic process that shows two distinct values. It models burst noise (also called popcorn noise or random telegraph signal). If the two possible values that a random variable can take are c 1 {\displaystyle c_{1}} and c 2 {\displaystyle c_{2}} , then the process can be described by the following master equations:
∂ t P ( c 1 , t | x , t 0 ) = − λ 1 P ( c 1 , t | x , t 0 ) + λ 2 P ( c 2 , t | x , t 0 ) {\displaystyle \partial _{t}P(c_{1},t|x,t_{0})=-\lambda _{1}P(c_{1},t|x,t_{0})+\lambda _{2}P(c_{2},t|x,t_{0})}
and
∂ t P ( c 2 , t | x , t 0 ) = λ 1 P ( c 1 , t | x , t 0 ) − λ 2 P ( c 2 , t | x , t 0 ) . {\displaystyle \partial _{t}P(c_{2},t|x,t_{0})=\lambda _{1}P(c_{1},t|x,t_{0})-\lambda _{2}P(c_{2},t|x,t_{0}).}
where λ 1 {\displaystyle \lambda _{1}} is the transition rate for going from state c 1 {\displaystyle c_{1}} to state c 2 {\displaystyle c_{2}} and λ 2 {\displaystyle \lambda _{2}} is the transition rate for going from going from state c 2 {\displaystyle c_{2}} to state c 1 {\displaystyle c_{1}} . The process is also known under the names Kac process (after mathematician Mark Kac), and dichotomous random process.
Solution The master equation is compactly written in a matrix form by introducing a vector P = [ P ( c 1 , t | x , t 0 ) , P ( c 2 , t | x , t 0 ) ] {\displaystyle \mathbf {P} =[P(c_{1},t|x,t_{0}),P(c_{2},t|x,t_{0})]} ,
d P d t = W P {\displaystyle {\frac {d\mathbf {P} }{dt}}=W\mathbf {P} }
where
W = ( − λ 1 λ 2 λ 1 − λ 2 ) {\displaystyle W={\begin{pmatrix}-\lambda _{1}&\lambda _{2}\\\lambda _{1}&-\lambda _{2}\end{pmatrix}}}
is the transition rate matrix. The formal solution is constructed from the initial condition P ( 0 ) {\displaystyle \mathbf {P} (0)} (that defines that at t = t 0 {\displaystyle t=t_{0}} , the state is x {\displaystyle x} ) by
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