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

### 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.

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*Mathematical notes of NEFU*, 26(3), pp. 31-46. doi: https://doi.org/10.25587/SVFU.2019.31.78.003.

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