Feynman-Kac and Feynman formulas for evolution pseudodifferential equations in superspace. (English) Zbl 1176.81051

Panyunin, N. M.

Russ. J. Math. Phys. 15, No. 4, 511-521 (2008).

Summary: In the paper, evolution pseudodifferential equations in the space of superanalytic functions \(\mathcal{A}(X)\) of an infinite-dimensional argument with symbols in the space \(\mathcal{F}(Y)\) of Fourier supertransforms of distributions on the dual superspace are considered. For these equations, the “weak” Cauchy problem is posed and the existence theorem for the solutions of this problem is proved. The main result of the paper is the theorem concerning the representation of solutions of the “weak” Cauchy problem by the Feynman path integral in the phase superspace (the Feynman-Kac formula). The Feynman integral is understood in the sequential sense. Thus, the Feynman formula becomes an immediate consequence of the Feynman-Kac formula.

MSC:

81Q30	Feynman integrals and graphs; applications of algebraic topology and algebraic geometry
35S30	Fourier integral operators applied to PDEs
46S60	Functional analysis on superspaces (supermanifolds) or graded spaces

References:

[1]	F. A. Berezin, Introduction to Algebra and Analysis with Anticommuting Variables (Moskov. Gos. Univ., Moscow, 1983); Introduction to Superanalysis (D. Reidel, Dordrecht-Boston, 1987).
[2]	V. S. Vladimirov and I.V. Volovich, ”Superanalysis. I. Differential Calculus,” Teoret. Mat. Fiz. 59(1), 3–27 (1984) [Theoret. and Math. Phys. 59 (1), 317–335 (1984)]. · Zbl 0552.46023
[3]	V. S. Vladimirov and I.V. Volovich, ”Superanalysis. II. Integral Calculus,” Teoret. Mat. Fiz. 60(2), 169–198 (1984) [Theoret. and Math. Phys. 60 (2), 743–765 (1984)]. · Zbl 0599.46068
[4]	A.Yu. Khrennikov, Superanalysis (Fizmatlit, Moscow, 2005; English transl. of the 1st ed.: Kluwer Academic Publishers, Dordrecht, 1999).
[5]	O. G. Smolyanov and E. T. Shavgulidze, ”The Fourier Transform and Pseudodifferential Operators in Superanalysis,” Dokl. Akad. Nauk SSSR 299(4), 816–820 (1988) [Soviet Math. Dokl. 37 (2), 476–481 (1988)]. · Zbl 0699.47036
[6]	A. P. Robertson and W. J. Robertson, Topological Vector Spaces (Cambridge University Press, New York, 1964; Mir, Moscow, 1967). · Zbl 0123.30202
[7]	A. Yu. Khrennikov, ”The Feynman Measure in a Phase Space and Symbols of Infinite-Dimensional Pseudodifferential Operators,” Mat. Zametki 37(5), 734–742 (1985) [Math. Notes 37 (5–6), 404–409 (1985)].
[8]	O. G. Smolyanov, A. G. Tokarev, and A. Truman, ”Hamiltonian Feynman Path Integrals via the Chernoff Formula,” J. Math. Phys. 43(10), 5161–5171 (2002). · Zbl 1060.58009 · doi:10.1063/1.1500422
[9]	O. G. Smolyanov and E. T. Shavgulidze, Path Integrals (Moskov. Gos. Univ., Moscow, 1990) [in Russian].
[10]	A. A. Slavnov,”Continual Integral in Perturbation Theory,” Teoret. Mat. Fiz. 22(2), 177–185 (1975) [in Russian].
[11]	P. R. Chernoff, ”Note on Product Formulas for Operator Semigroups,” J. Funct. Anal. 2, 238–242 (1968). · Zbl 0157.21501 · doi:10.1016/0022-1236(68)90020-7
[12]	N. M. Panyunin, ”Fourier Transform of Supermeasures,” Russ. J. Math. Phys. 14(4), 501–504 (2007). · Zbl 1180.46058 · doi:10.1134/S1061920807040176