In our paper, we construct an analog of probabilistic representation of the solution of the Cauchy problem for the equation \( \frac{{\partial u}}{{\partial t}}+\frac{{{\sigma^2}}}{2}\frac{{{\partial^2}u}}{{\partial {x^2}}}+f(x)u=0, \) where σ is a complex number. Bibliography: 7 titles.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
N V. Smorodina and M. M. Faddeev, “Probabilistic representation of solutions for a class of evolution equations.” Zap. Nauchn. Semin. POMI, 384, 238–266 (2010).
A. D. Ventzel, Course of Theory of Random Processes [in Russian], Moscow (1975).
H. Doss, “Sur une résolution stochastique de l’equation de Schrödinger à coefficients analytiques,” Commun. Math. Phys., 73, 247–264 (1980).
I. M. Gelfand and G. E. Shilov, Generalized Functions. Properties and Operations [in Russian]. Moscow (1958).
Yu. L. Daletskii and S. V. Fomin, Measures and Differential Equations in Functional Spaces [in Russian], Moscow (1983).
A. V. Skorokhod, Random Processes With Independent Increments [in Russian], Moscow (1986).
M. Reed and B. Simon, Methods of Modern Mathematical Physics [Russian translation]. Moscow (1978).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 396, 2012, pp. 111–143.
Rights and permissions
About this article
Cite this article
Ibragimov, I.A., Smorodina, N.V. & Faddeev, M.M. Probabilistic approximation of solutions of the cauchy problem for some evolution equations. J Math Sci 188, 700–716 (2013). https://doi.org/10.1007/s10958-013-1161-8
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-013-1161-8