**T.I. Nasirova, E.A. Hajiyev, G.R. Gasimov, U.D. Idrisova, T.A. Aliyev**

Mangeron's equation in the semi-Markov random walk process

Mangeron's equation in the semi-Markov random walk process

This paper considers the sequence {ξ_k^+,η_k^+,ξ_k^- }_(k=1)^∞ of independent identically distributed, positive, independent random variables and the sequence { η_k^- }_(k=1)^∞ of negative random variables. On the basis of these random variables, a semi-Markov random walk process with a delaying screen at zero is constructed, and an integral equation for the conditional distribution R(├ t,x┤|z,h) of this process is found using the formula of total probability. In the class of distributions decreasing exponentially fast, using the method of successive Laplace integral transforms in time t and Laplace-Stiltes in phase x, this integral equation is reduced to a partial differential equation – to the fourth-order Mangeron equation. Thus, it is proved that the double integral image R ̃ ̃(├ θ,α┤|z,h) of the conditional distribution of the constructed semi-Markov random walk process is a solution to the obtained Mangeron equation.

**Keywords:**Total probability formula, Laplace-Stieltjes transform, Independent random variables, Distribution, Markov quantities