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Teor. Veroyatnost. i Primenen., 1964, Volume 9, Issue 3, Pages 466–491 (Mi tvp393)  

This article is cited in 2 scientific papers (total in 3 papers)

On Brownian Motion Equations

A. M. Il'ina, R. Z. Has'minskiĭb

a Sverdlovsk
b Moscow

Abstract: A study is made of the relationships between the different descriptions of Brownian motion expressed as an integro-differential equation of Boltzmann type, as a Langevin equation and a partial differential equation corresponding to it, and as Fokker–Plank–Kolmogorov equations. Special attention is devoted to the relationships between the last two descriptions. Let the mass of a particle be $m$, the temperature of the medium $T$, the viscosity of the medium $A$, and let the intensity of the power field in $x$ be $F(x)$. Then the Brownian motion equation in the phase space $(x,y=\dot x)$ has the form (3.4). In the appendix to this paper the existence of the Green function for equation (3.4) is proved. An asymptotic series is obtained as the solution to the Cauchy problem for equation (3.4) for $\varepsilon={m/{A\ll 1}}$. In particular it is proved that the zero term of this asymptotic series for $t\gg\varepsilon$ is the solution to the Cauchy problem for equation (4.13) under suitable initial conditions.

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English version:
Theory of Probability and its Applications, 1964, 9:3, 421–444

Bibliographic databases:

Received: 11.04.1963

Citation: A. M. Il'in, R. Z. Has'minskiǐ, “On Brownian Motion Equations”, Teor. Veroyatnost. i Primenen., 9:3 (1964), 466–491; Theory Probab. Appl., 9:3 (1964), 421–444

Citation in format AMSBIB
\Bibitem{IliKha64}
\by A.~M.~Il'in, R.~Z.~Has'minski{\v\i}
\paper On Brownian Motion Equations
\jour Teor. Veroyatnost. i Primenen.
\yr 1964
\vol 9
\issue 3
\pages 466--491
\mathnet{http://mi.mathnet.ru/tvp393}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=168018}
\zmath{https://zbmath.org/?q=an:0134.34303}
\transl
\jour Theory Probab. Appl.
\yr 1964
\vol 9
\issue 3
\pages 421--444
\crossref{https://doi.org/10.1137/1109058}


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    This publication is cited in the following articles:
    1. V. M. Zatsepin, “Time correlation functions of one-dimensional rotational Brownian motion in an $n$-fold periodic potential”, Theoret. and Math. Phys., 33:3 (1977), 1099–1105  mathnet  crossref  mathscinet  zmath
    2. V. M. Babich, L. A. Kalyakin, M. D. Ramazanov, N. Kh. Rozov, “Arlen Mikhailovich Il'in (on the occasion of the 70th anniversary)”, Proc. Steklov Inst. Math. (Suppl.), 2003no. , suppl. 1, S1–S7  mathnet  mathscinet  zmath  elib
    3. “Arlen Mikhailovich Ilin (k vosmidesyatiletiyu so dnya rozhdeniya)”, Ufimsk. matem. zhurn., 4:2 (2012), 3–12  mathnet  mathscinet
  • Теория вероятностей и ее применения Theory of Probability and its Applications
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