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首页> 外文期刊>Doklady. Mathematics >Feynman Formulas for Evolution Equations with Levy Laplacians on Infinlte-Dimensfonal Manifolds
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Feynman Formulas for Evolution Equations with Levy Laplacians on Infinlte-Dimensfonal Manifolds

机译：无穷二维流形上带有征费拉普拉斯算子的演化方程的费曼公式

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摘要

A Feynman formula is a representation of the solution to the Cauchy problem for an evolution partial differential (or pseudodifferential) equation in terms of the limit of a sequence of multiple integrals with multiplicities tending to infinity. The integrands are products of the initial condition and Gaussian (or complex Gaussian) exponentials1 [5]. In this paper, we obtain Feynman formulas for the solutions to the Cauchy problems for the Schrodinger equation and the heat equation with Levy Laplacian on the infinite-dimensional manifold of mappings from a closed real interval to a Riemannian manifold. The definition of the Levi Laplacian acting on functions on such a manifold is obtained by combining the methods of papers [3] and [7]. In the former, Levi Laplacians in the space of functions on an infinite- imensional vector space were considered, and in the latter, Volterra Laplacians in the space of functions on the above infinite-dimensional manifold were examined. This definition of a Levi Laplacian is equivalent to that given in [2], but it is better adapted for derivation of Feynman formulas.
机译：Feynman公式用多重积分的序列的极限趋于无穷大来表示演化偏微分（或伪微分）方程的柯西问题的解。整数是初始条件和高斯（或复数高斯）指数的乘积[5]。在本文中，我们获得了从封闭实区间到黎曼流形的无穷维流形上Schrodinger方程和Levy Laplacian热方程的Cauchy问题解的费曼公式。通过结合论文[3]和[7]的方法，可以得出作用在这种流形上的李维·拉普拉斯算子的定义。在前者中，考虑了在无穷维矢量空间上的函数空间中的Levi Laplacians，在后者中，研究了在上述无穷维流形上的函数空间中的Volterra Laplacian。 Levi Laplacian的此定义与[2]中给出的定义相同，但更适合于推导Feynman公式。

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