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Teoriya Veroyatnostei i ee Primeneniya, 1967, Volume 12, Issue 3, Pages 540–547 (Mi tvp736)  

This article is cited in 15 scientific papers (total in 15 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,\dots,x_1^n,x_2^1,\dots,x_2^n,\dots,x_N^1,\dots,x_N^n,t)\frac{\partial^2u}{\partial x_1^i\partial x_1^j}+ \\ +\sum_{i=1}^na_i(x_1^1,\dots,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}+\dots \\ \dots+\sum_{i=1}^nx_{N-1}^i\frac{\partial u}{\partial x_N^i}+a(x_1^1,\dots,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.
Received: 05.05.1966
English version:
Theory of Probability and its Applications, 1967, Volume 12, Issue 3, Pages 490–496
DOI: https://doi.org/10.1137/1112059
Bibliographic databases:
Document Type: Article
Language: Russian
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://mathscinet.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}
Linking options:
  • https://www.mathnet.ru/eng/tvp736
  • https://www.mathnet.ru/eng/tvp/v12/i3/p540
  • This publication is cited in the following 15 articles:
    1. Cloud Makasu, “Remarks on a controlled degenerate diffusion process”, European Journal of Control, 77 (2024), 100986  crossref
    2. Tommaso Barbieri, “On Kolmogorov Fokker Planck operators with linear drift and time dependent measurable coefficients”, MINE, 6:2 (2024), 238  crossref
    3. Anceschi F. Muzzioli S. Polidoro S., “Existence of a Fundamental Solution of Partial Differential Equations Associated to Asian Options”, Nonlinear Anal.-Real World Appl., 62 (2021), 103373  crossref  isi
    4. Marco Bramanti, Sergio Polidoro, “Fundamental solutions for Kolmogorov-Fokker-Planck operators with time-depending measurable coefficients”, Mathematics in Engineering, 2:4 (2020), 734  crossref
    5. I. P. Medynsky, “On properties of solutions for Fokker-Planck-Kolmogorov equations”, Math. Model. Comput., 7:1 (2020), 158  crossref
    6. A. A. Kozhina, “Weak error for the Euler scheme approximation of degenerate diffusions with nonsmooth coefficients”, J. Math. Sci., 254:4 (2021), 510–531  mathnet  crossref
    7. S. D. Ivasishen, I. P. Medynsky, “The Fokker–Planck–Kolmogorov equations for some degenerate diffusion processes”, Theory Stoch. Process., 16(32):1 (2010), 57–66  mathnet  mathscinet  zmath
    8. A.J. Koerber, J.R. King, P. Williams, “Deterministic and stochastic modelling of endosome escape by Staphylococcus aureus: ?quorum? sensing by a single bacterium”, J. Math. Biol., 50:4 (2005), 440  crossref
    9. M.A. Kouritzin, “On exact filters for continuous signals with discrete observations”, IEEE Trans. Automat. Contr., 43:5 (1998), 709  crossref
    10. 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
    11. S. A. Molchanov, “The structure of eigenfunctions of one-dimensional unordered structures”, Math. USSR-Izv., 12:1 (1978), 69–101  mathnet  crossref  mathscinet  zmath
    12. Avner Friedman, Lecture Notes in Mathematics, 415, Ordinary and Partial Differential Equations, 1974, 144  crossref
    13. Makiko Nisio, Lecture Notes in Mathematics, 330, Proceedings of the Second Japan-USSR Symposium on Probability Theory, 1973, 316  crossref
    14. V. N. Dubrovsky, “On small random perturbations of a second order differential equation”, Theory Probab. Appl., 18:3 (1974), 476–485  mathnet  mathnet  crossref
    15. M. I. Freidlin, “On the smoothness of solutions of degenerate elliptic equations”, Math. USSR-Izv., 2:6 (1968), 1337–1359  mathnet  crossref  mathscinet  zmath
    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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