Abstract
In this paper, we show how to construct an asymptotic representation of the fundamental solution to the Cauchy problem for degenerate linear parabolic equations.
DOI 10.1134/S1061920821020047
Similar content being viewed by others
References
V. G. Danilov and S. M. Frolovitchev, “Exact Asymptotics of the Density of the Transition Probability for Discontinuous Markov Processes”, Math. Nachr., 215 (2000), 55–90.
V. G. Danilov, “A Representation of the Delta Function via Creation Operators and Gaussian Exponentials and Multiplicative Fundamental Solution Asymptotics for Some Parabolic Pseudodifferential Equations”, Russ. J. Math. Phys., 3:1 (1995), 25–40.
V. G. Danilov and S. M. Frolovitchev, “Exact Asymptotics of the Density of the Transition Probability for Discontinuous Markov Processes”, Math. Nachr., 215 (2000), 55–90.
M. V. Fedoryuk and V. P. Maslov, Semiclassical Approximation in Quantum Mechanics, Reidel, Dordrecht, 1976.
L. HHörmander, “Fourier Integral Operators”, Acta Math., 127 (1971), 79–183.
Yu. I. Kifer, “Some Results Concerning Small Stochastic Perturbations of Dynamical Systems”, Teor. Veroyatnost. i Primenen. (Theory Probab. Appl.) XIX, 3 (1974), 514–532.
Yu. I. Kifer, “On the Asymptotics of the Density of the Transition Probability for Processes with Small Diffusion”, Teor. Veroyatnost. i Primenen. (Theory Probab. Appl.) XXI, 3 (1976), 527–536.
V. P. Maslov, “Global Exponential Asymptotics of the Solutions of the Tunnel Equations and the Large Deviations Problem”, Tr. Mat. Inst. Steklov, Moscow, 163:163 (1984), 150–180.
V. P. Maslov, Asymptotic Methods and the Perturbation Theory, Nauka, Moscow, 1988 [in Russian].
V. P. Maslov and V. E. Nazaikinskii, “Tunnel Canonical Operator in Thermodynamics”, Funct. Anal. Appl., 40 (2006), 173–187.
W. Matsumoto, “Local Solvability of a Class of Partial Differential Equations with Multiple Characteristics”, Proc. Japan Acad., 51:3 (1975), 151–155.
Acknowledgments
Finally, we would like to emphasize that the above approach arose during very pleasant and useful discussions with our teacher V.P. Maslov and one of the authors keeps these unforgettable discussions in his memory.
Funding
The study was implemented in the framework of the Basic Research Program at the National Research University Higher School of Economics (HSE University) in 2021.
Author information
Authors and Affiliations
Corresponding authors
Rights and permissions
About this article
Cite this article
Danilov, V.G., Rakhel, M.A. Development of Maslov’s Approach to the Construction of Nonoscillating WKB-Type Solutions. Russ. J. Math. Phys. 28, 179–187 (2021). https://doi.org/10.1134/S1061920821020047
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1061920821020047