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Teoriya Veroyatnostei i ee Primeneniya, 1964, Volume 9, Issue 3, Pages 466–491 (Mi tvp393)  

This article is cited in 34 scientific papers (total in 35 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.
Received: 11.04.1963
English version:
Theory of Probability and its Applications, 1964, Volume 9, Issue 3, Pages 421–444
DOI: https://doi.org/10.1137/1109058
Bibliographic databases:
Language: Russian
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://mathscinet.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 35 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Теория вероятностей и ее применения Theory of Probability and its Applications
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