×

Evolution processes and the Feynman-Kac formula. (English) Zbl 0844.60027

Mathematics and its Applications (Dordrecht). 353. Dordrecht: Kluwer Academic Publishers. ix, 235 p. (1996).
In the late forties R. Feynman proposed to represent the dynamics generated by the perturbed Hamiltonian as a path integral with respect to the “process” corresponding to the free dynamics. This idea turned out to be very successful in the area of Markov processes, where it has led to the Feynman-Kac formula.
The aim of the book is to investigate a broad class of evolutions for which an appropriate version of the Feynman-Kac formula can be derived. The problem of path integral is intimately related to the theory of vector measures which is the main tool in this book. In particular the theory of bilinear integration and integration with respect to unbounded set functions is developed. Then the author introduces the concept of evolution process associated with the operator-valued measure which enjoys the Markov property. For any semigroup of operators acting on \(E\) and any spectral measure (corresponding to possible observables) such a process with the state space \(E\) can be constructed. Then the operator-valued multiplicative functional is defined and it is shown that in such a framework the Feynman-Kac formula still holds provided the underlying process is associated to a bounded operator-valued measure. The case of unbounded measure, which is closer to the original Feynman idea, is much harder. To tackle this problem the author develops the theory of integration for unbounded set functions by means of regularizing measures. Finally, this theory is applied to the Schrödinger and Dirac equations for which the Feynman-Kac formula is obtained and to systems of differential equations of the first order.
A sample of topics discussed in the book: Markov evolution processes, A general Feynman-Kac formula for bounded processes, Bilinear integration, Random evolutions, A noncommutative Feynman-Kac formula, Operator valued transition functions, Semigroups on \(L^\infty\) with a bounded generator, The direct sum of dynamical systems, Representation of evolutions, Integration with respect to unbounded set functions, The Schrödinger process and its Feynman representation, The radial Dirac process, Construction of cut-off measures, The Feynman representation for the radial Dirac process.

MSC:

60Hxx Stochastic analysis
60-02 Research exposition (monographs, survey articles) pertaining to probability theory
28A05 Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets
46G10 Vector-valued measures and integration
81Q30 Feynman integrals and graphs; applications of algebraic topology and algebraic geometry
PDFBibTeX XMLCite