8 research outputs found
Solution to a parabolic differential equation in Hilbert space via Feynman formula - parts I and II
A parabolic partial differential equation is considered,
where is a linear second-order differential operator with time-independent
coefficients, which may depend on . We assume that the spatial coordinate
belongs to a finite- or infinite-dimensional real separable Hilbert space
.
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 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 . Moreover, this solution depends
continuously on the initial condition. In the case where the coefficient of the
first-derivative term in 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.Comment: This is a more or less stable version of the tex