×

zbMATH — the first resource for mathematics

Approximation of the evolution operator by expectations of functionals of sums of independent random variables. (English. Russian original) Zbl 07062743
Theory Probab. Appl. 64, No. 1, 12-26 (2019); translation from Teor. Veroyatn. Primen. 64, No. 1, 17-35 (2019).
MSC:
60 Probability theory and stochastic processes
68 Computer science
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] A. D. Wentzell, A Course in the Theory of Stochastic Processes, McGraw-Hill International Book Co., New York, 1981.
[2] N. Dunford and J. T. Schwartz, Linear Operators, Vol. I: General Theory, Pure Appl. Math. 7, Interscience Publishers, Inc., New York, 1958.
[3] H. Doss, Sur une resolution stochastique de l’equation de Schrödinger à coefficients analytiques, Comm. Math. Phys., 73 (1980), pp. 247–264. · Zbl 0427.60099
[4] J. Glimm and A. Jaffe, Quantum Physics. A Functional Integral Point of View, 2nd ed., Springer-Verlag, New York, 1987.
[5] I. A. Ibragimov, N. V. Smorodina, and M. M. Faddeev, On a limit theorem related to probabilistic representation of solution to the Cauchy problem for the Schrödinger equation, J. Math. Sci. (N.Y.), 229 (2018), pp. 702–713.
[6] M. M. Faddeev, I. A. Ibragimov, and N. V. Smorodina, A stochastic interpretation of the Cauchy problem solution for the equation \(\partial_t u=(\sigma^2/2)\Delta u +V(x)u\) with complex \(\sigma\), Markov Process. Related Fields, 22 (2016), pp. 203–226. · Zbl 1366.60089
[7] I. A. Ibragimov, N. V. Smorodina, and M. M. Faddeev, Probabilistic approximation of the evolution operator, Funct. Anal. Appl., 52 (2018), pp. 101–112. · Zbl 06951618
[8] T. Kato, Perturbation Theory for Linear Operators, Grundlehren Math. Wiss. 132, Springer-Verlag, New York, 1966.
[9] J. F. C. Kingman, Poisson Processes, Oxford Stud. Probab. 3, The Clarendon Press, Oxford Univ. Press, New York, 1993.
[10] M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. II: Fourier Analysis, Self-Adjointness, Academic Press, New York, 1975. · Zbl 0308.47002
[11] O. G. Smolyanov and E. T. Shavgulidze, Path Integrals, 2nd ed., Lenand, Moscow, 2015 (in Russian).
[12] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Math. Ser. 30, Princeton Univ. Press, Princeton, NJ, 1970.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.