A Feynman-Kac solution to a random impulsive equation of Schrödinger type. (English) Zbl 1246.28008

Bonotto, E. M.; Federson, M.; Muldowney, P.

Real Anal. Exch. 36(2010-2011), No. 1, 107-148 (2011).

If a force is applied to a particle undergoing Brownian motion, the resulting motion has a state function which satisfies a diffusion or Schrödinger-type equation. In this paper, the authors consider a process in which Brownian motion is replaced by a process which has Brownian transitions at all times other than random times at which the transitions have an additional “impulsive” displacement.
The purpose of this paper is to examine the relationship between discontinuities in the state function which characterizes the diffusion, and impulsive changes in the underlying diffusion itself. They use a Feynman-Kac formulation to show the connection between these two classes of discontinuities. The method of analysis is based on the generalized Riemann approach of Henstock. In effect, the Feynman-Kac formulation of the problem is a generalized Riemann (or Henstock) integral.

Reviewer: Kun Soo Chang (Seoul)

MSC:

28C20	Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.)
35R12	Impulsive partial differential equations
46G12	Measures and integration on abstract linear spaces

Keywords:

Henstock integral; Feynman-Kac formula; partial differential equations; impulse; Brownian motion