Abstract
A parabolic partial differential equation u′ t (t, x) = Lu(t, x) is considered, where L is a linear second-order differential operator with time-independent coefficients, which may depend on x . We assume that the spatial coordinate x belongs to a finiteor infinite-dimensional real separable Hilbert space H . Assuming the existence of a strongly continuous resolving semigroup for this equation, we construct a representation of this semigroup by a Feynman formula, i.e. we write it in the form of the limit of a multiple integral over H as the multiplicity of the integral tends to infinity. This representation gives a unique solution to the Cauchy problem in the uniform closure of the set of smooth cylindrical functions on H . Moreover, this solution depends continuously on the initial condition. In the case where the coefficient of the first-derivative term in L vanishes we prove that the strongly continuous resolving semigroup exists (this implies the existence of the unique solution to the Cauchy problem in the class mentioned above) and that the solution to the Cauchy problem depends continuously on the coefficients of the equation. The article is published in the author’s wording.
Highlights
Representation of a function by the limit of a multiple integral as multiplicity tends to infinity is called a Feynman formula, after R.P
It is important to note that Feynman formulas are closely related to Feynman-Kac formulas [30], the latter will not be studied in the present article
The solution is given in the form of a convolution with the Gaussian measure, the existence of the resolving semigroup is proved
Summary
Representation of a function by the limit of a multiple integral as multiplicity tends to infinity is called a Feynman formula, after R.P. In [22] a solution to a heat equation in Hilbert space without the terms of the first and zero order is discussed, the coefficient of the second-derivative term is constant. The solution is given in the form of a convolution with the Gaussian measure (analogous to the finite dimensional equation with constant coefficients), the existence of the resolving semigroup is proved. In [28], for a class of equations in an infinite-dimensional space, with a variable coefficient at the highest derivative (but without first- and zero-order derivatives’ terms), a Feynman formula was obtained and the existence of resolving semigroup was proven. In spaces over the field of p-adic numbers, Feynman and Feynman-Kac formulas for the solutions of the Cauchy problem for evolutionary equations were given in [11, 12]. The article may be used as a very short introduction to analysis in Hilbert space and to the applications of C0-semigroup theory in solving evolutionary PDEs
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