Feynman Formulas and Path Integrals for Evolution Equations with the Vladimirov Operator

O. G. Smolyanov ; N. N. Shamarov

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Selected topics of mathematical physics and $p$-adic analysis, Tome 265 (2009), pp. 229-240

Résumé

We obtain Feynman formulas in the momentum space and Feynman–Kac formulas in the momentum and phase spaces for a $\mathfrak p$-adic analog of the heat equation in which the role of the Laplace operator is played by the Vladimirov operator. We also present the Feynman and Feynman–Kac formulas in the configuration space that have been proved in our previous papers under additional constraints. In all these formulas, integration is performed with respect to countably additive measures. The technique developed in the paper is fundamentally different from that used by the authors when studying path integrals in configuration spaces. In particular, the paper extensively uses the infinite-dimensional Fourier transform.