ON THE SOLUTION OF THE STOCHASTIC HELMHOLTZ PROBLEM BY THE METHOD OF PHASE SPACE’S TRANSFORMATION BY VELOCITIES

Abstract

The classical Helmholtz problem is the problem of constructing equivalent Lagrange-shaped differential equations from given second-order ordinary differential equations. In this paper, stochastic equations of the Lagrangian structure, equivalent in distribution, are constructed using the second-order Ito equations. The phase space transformation method is used to solve the stochastic Helmholtz problem. The results obtained are illustrated by two examples.  The solvability analysis of the stochastic Helmholtz problem is carried out in the class of stochastic differential equations equivalent in distribution, in contrast to the work [Tleubergenov, M. I., Azhymbaev, D. T. On the Solvability of Stochastic Helmholtz Problem. J. Math. Sci. Vol. 253. – P. 297–305 (2021)], in which the Helmholtz problem is solved by the method of addi-tional variables in the class of stochastic differential equations equivalent almost surely (a.s.).  The study of the stochastic Helmholtz problem in the class of equations equivalent in distribution allows us to significantly expand the domain of solvability of this problem due to the possibility of using the phase space transformation method in this class.