# Peculiarities of the dynamics of a brownian particle with random disturbances orthogonal to its speed

• Dubko Valerii A., doobko2017@ukr.net Kyiv National University of Technologies and Design, Education and Scientific Institute of Modern Learning Technologies, Department of Higher Mathematics, 2, building 4, Nemyrovych-Danchenko Street, Kyiv 01011, Ukraine
Keywords: Langevin equation, orthogonal perturbations, diffusion approximation, wave equation, first integral

### Abstract

The classical diffusion equations are based on the assumption that the velocities of a Brownian particle can take arbitrarily large values. In this article, it is shown that for solving the Langevin equations when random influences are orthogonal to the particle velocity there might exist an attractive surface for velocity, despite the fact that the Wiener process is a process that takes arbitrarily large values. Unlike the previous articles, here we construct an equation for determining the probability density of the distribution of particles in the coordinate space taking the initial direction of velocity into account. It is shown that small influences with a certain agreement of the coefficients in the initial stochastic equation lead to a description of the moving particle based on the wave equations with constant speed. The considered equations do not exhaust the class of models when the perturbations are orthogonal to the vector component of the solution. An extended class of stochastic equations with orthogonal perturbations was considered in previous works of the author, in particular, for n-dimensional processes, in connection with the development of the theory of first integrals for stochastic systems.

### References

[1]
Langevin P., Selected Works [in Russian], Izdat. Akad. Nauk SSSR, Moscow (1960); Nauka, Moscow (1964).

[2]
Skorokhod A. V., “Stochastic equations of a system of many particles [in Russian],” in: Mathematical Methods in Biology, pp. 33–53, Nauk. Dumka, Kiev (1977).

[3]
Kac M., Several Probabilistic Problems of Physics and Mathematics [in Russian], Nauka, Moscow (1967).

[4]
Kac M., “A stochastic model related to the telegrapher’s equation,” Rocky Mount. J. Math., 4, No. 3, 497–510 (1974).

[5]
Kolesnik A. D., “On a model of Markov random evolution on the plane [in Russian],” in: Analytical Methods for Studying the Evolution of Stochastic Systems, Collect. Sci. Works, pp. 55–61, Inst. Mat., Akad. Nauk Ukr. SSR, Kiev (1989).

[6]
Kolesnik A. D. and Turbin A. F., “Symmetric Markovian random evolution on the plane [in Russian],” preprint 90.12, 39 p., Inst. Mat. Akad. Nauk Ukr. SSR, Kiev (1990),

[7]
Kolesnik A. D. and Turbin A. F., “Symmetric random evolution in R2 [in Russian],” Dokl. Akad. Nauk Ukr. SSR, 2, 10–11 (1990).

[8]
Kolesnik A. D. and Turbin A. F., “Infinitesimal hyperbolic operator of Markov random evolution in R2 [in Russian],” Dokl. Akad. Nauk Ukr. SSR, 1, 11–14 (1991).

[9]
Orsingher E., “A planar random motion governed by the two-dimensional telegraph equation,” J. Appl. Probab., 23, 385–397 (1986).

[10]
Orsingher E., “Probability law, flow rate, maximum distribution of wave governed by motions and their laws with Kirchoff’s laws,” Stoch. Process. Appl., 34, 49–66 (1990).

[11]
Orsingher E. and Kolesnik A., “The exact distribution in the model of random motion on a plane controlled by a fourth-order hyperbolic equation [in Russian],” Theory Probab. Appl., 41, No. 2, 451–459 (1996).

[12]
Turbin A. F., “The one-dimensional process of Brownian motion is an alternative to the model of A. Einstein–N. Wiener–P. Levy [in Russian],” Fraktal. Anal. Pitan., 2, 47–60 (1998).

[13]
Sevilla F. J. and Nava L. A. G., “Theory of diffusion of active particles that move at constant speed in two dimensions,” Phys. Rev., E 90, 022130, (publ. 25 Aug. 2014).

[14]
Dubko V. A., “The first integral of the system of stochastic differential equations [in Russian],” preprint 78.27, 28 p., Inst. Mat. Akad. Nauk Ukr. SSR, Kiev (1978).

[15]
Dubko V. A., Questions of the Theory and Application of Stochastic Differential Equations [in Russian], Dal’nevost. Otd. Akad. Nauk SSSR, Vladivostok (1989).

[16]
Gikhman I. I. and Skorokhod A. V., Stochastic Differential Equations [in Russian], Nauk. Dumka, Kiev (1968).

[17]
Skorohod A. V., “On the averaging of stochastic equations of mathematical physics [in Russian],” in: Problems of Asymptotic Theory of Nonlinear Oscillations, pp. 196–208, Nauk. Dumka, Kiev (1977).

[18]
Dubko V. A., “Lowering the order of a system of stochastic differential equations with a small parameter with the highest derivative [in Russian],” Teor. Sluch. Protsess., 8, 35–41 (1980).

[19]
Dubko V. A., “Modeling the dynamics of real processes [in Russian],” in: Dokl. III Mezhdunar. Konf. "Ecological and Geographical Problems of Environmental Management in Oil and Gas Regions", pp. 16–20. NGGU, Nizhnevartovsk (2006).

[20]
Karachanskaya E. V. and Petrova A. P., “Non-random functions and solutions of Langevintype stochastic differential equations [in Russian],” Mat. Zamet. SVFU, 23, No. 3, 55–69 (2016).
How to Cite
Dubko, V. (2019) “Peculiarities of the dynamics of a brownian particle with random disturbances orthogonal to its speed”, Mathematical notes of NEFU, 26(3), pp. 31-46. doi: https://doi.org/10.25587/SVFU.2019.31.78.003.
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Mathematics