### Speaker

Dr
Dmitry Kulyabov
(PFUR & JINR)

### Description

When modeling different physical and technical systems, they can often be modeled in the form of one-step processes. Our group has been developing a formalism of stochastization of one-step processes for quite a long time. We investigated a variety of representations of both the one-step processes, and methods of their stochastization. We have considered representations in the state vectors (combinatorial approach) and in the occupation numbers (operator approach) [1]. With stochastization of systems with control, we use a geometric approach to control theory. It would be useful to consider the geometric approach also to the methods of stochastization of one-step processes.
We have considered various variants of geometrization of the process of stochastization of one-step processes and stochastic differential equations. Approaches were considered both on the basis of Riemannian quadratic metrics [2-3] and on the basis of a more general approach of Finsler geometry [4-8].
Different approaches to geometrization of stochastic systems are considered in the paper and comparison with other methodological approaches is made.
The work is partially supported by RFBR grants No's 15-07-08795 and 16-07-00556. Also the publication was financially supported by the Ministry of Education and Science of the Russian Federation (the Agreement No 02.A03.21.0008).
References
1. M. Hnatič, E. G. Eferina, A. V. Korolkova, D. S. Kulyabov, L. A. Sevastyanov, Operator Approach to the Master Equation for the One-Step Process, EPJ Web of Conferences 108 (2016) 02027. arXiv:1603.02205, doi:10.1051/epjconf/201610802027.
2. Graham, R., 1977. Covariant formulation of non-equilibrium statistical thermodynamics. Zeitschrift für Physik B Condensed Matter and Quanta 26, 397–405. doi:10.1007/BF01570750
3. Graham, R., 1977. Path integral formulation of general diffusion processes. Zeitschrift für Physik B Condensed Matter and Quanta 26, 281–290. doi:10.1007/BF01312935
4. Asanov, G.S., 1985. Finsler Geometry, Relativity and Gauge Theories. Springer Netherlands, Dordrecht. doi:10.1007/978-94-009-5329-1
5. Rund, H., 1959. The Differential Geometry of Finsler Spaces. Springer Berlin Heidelberg, Berlin, Heidelberg. doi:10.1007/978-3-642-51610-8
6. Antonelli, P.L., Ingarden, R.S., Matsumoto, M., 1993. The Theory of Sprays and Finsler Spaces with Applications in Physics and Biology, Fundamental Theories of Physics. Volume 58. Springer Netherlands, Dordrecht. doi:10.1007/978-94-015-8194-3
7. Antonelli, P.L., Miron, R. (Eds.), 1996. Lagrange and Finsler Geometry, Fundamental Theories of Physics . Volume 76. Springer Netherlands, Dordrecht. doi:10.1007/978-94-015-8650-4
8. Antonelli, P.L., Zastawniak, T.J., 1999. Fundamentals of Finslerian Diffusion with Applications, Fundamental Theories of Physics. Volume 101. Springer Netherlands, Dordrecht. doi:10.1007/978-94-011-4824-5

### Primary author

Dr
Dmitry Kulyabov
(PFUR & JINR)

### Co-authors

Anna Korolkova
(Peoples' Friendship University of Russia)
Ekaterina Eferina
(Peoples’ Friendship University of Russia (RUDN University))
Prof.
Leonid Sevastyanov
(PFUR)