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Teor. Veroyatnost. i Primenen., 1967, Volume 12, Issue 3, Pages 540–547 (Mi tvp736)  

This article is cited in 5 scientific papers (total in 5 papers)

Short Communications

On a class degenerative diffusion processes

I. M. Sonin

Moscow

Abstract: It is proved that the degenerative diffusion processes with the characteristic operators reducing after a change of variables to the form
\begin{gather*} Lu=\frac12\sum_{ij=1}^na_{ij}(x_1^1,…,x_1^n,x_2^1,…,x_2^n,…,x_N^1,…,x_N^n,t)\frac{\partial^2u}{\partial x_1^i\partial x_1^j}+
+\sum_{i=1}^na_i(x_1^1,…,x_N^n,t)\frac{\partial u}{\partial x_1^i}+\sum_{i=1}^nx_1^i\frac{\partial u}{\partial x_2^i}+\sum_{i=1}^nx_2^i\frac{\partial u}{\partial x_3^i}+…
…+\sum_{i=1}^nx_{N-1}^i\frac{\partial u}{\partial x_N^i}+a(x_1^1,…,x_N^n,t)u \end{gather*}
have smooth densities. The proof is carried out by constructing the fundamental solution of the corresponding parabolic equation. For this the classical method of Lévy with some modifications is used.

Full text: PDF file (449 kB)

English version:
Theory of Probability and its Applications, 1967, 12:3, 490–496

Bibliographic databases:

Received: 05.05.1966

Citation: I. M. Sonin, “On a class degenerative diffusion processes”, Teor. Veroyatnost. i Primenen., 12:3 (1967), 540–547; Theory Probab. Appl., 12:3 (1967), 490–496

Citation in format AMSBIB
\Bibitem{Son67}
\by I.~M.~Sonin
\paper On a~class degenerative diffusion processes
\jour Teor. Veroyatnost. i Primenen.
\yr 1967
\vol 12
\issue 3
\pages 540--547
\mathnet{http://mi.mathnet.ru/tvp736}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=215367}
\zmath{https://zbmath.org/?q=an:0183.19901}
\transl
\jour Theory Probab. Appl.
\yr 1967
\vol 12
\issue 3
\pages 490--496
\crossref{https://doi.org/10.1137/1112059}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. M. I. Freidlin, “On the smoothness of solutions of degenerate elliptic equations”, Math. USSR-Izv., 2:6 (1968), 1337–1359  mathnet  crossref  mathscinet  zmath
    2. S. A. Molchanov, “The structure of eigenfunctions of one-dimensional unordered structures”, Math. USSR-Izv., 12:1 (1978), 69–101  mathnet  crossref  mathscinet  zmath
    3. Nguyen Dinh Cong, “On the stochastic stability of the Lyapunov exponents of equations of arbitrary order”, Math. USSR-Sb., 60:1 (1988), 217–235  mathnet  crossref  mathscinet  zmath
    4. S. D. Ivasishen, I. P. Medynsky, “The FokkerPlanckKolmogorov equations for some degenerate diffusion processes”, Theory Stoch. Process., 16(32):1 (2010), 57–66  mathnet  mathscinet  zmath
    5. A. Kozhina, “Slabaya oshibka skhemy Eilera dlya vyrozhdennykh diffuzii s negladkimi koeffitsientami”, Fundament. i prikl. matem., 22:3 (2018), 91–118  mathnet
  •     Theory of Probability and its Applications
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