Evolution Processes and the Feynman-Kac Formula by Brian Jefferies

By Brian Jefferies

This ebook is an outgrowth of rules originating from 1. Kluvanek. regrettably, Professor Kluvanek didn't stay to give a contribution to the undertaking of writing up in a scientific shape, the circle of principles to which the current paintings is dedicated. it truly is most likely that together with his enter, the process and parts of emphasis of the ensuing exposition might were particularly assorted from what we've right here. however, the stamp of Kluvanek's inspiration and philosophy (but now not unavoidably his approval) abounds all through this publication. even though the name provides no indication, integration concept in vector areas is a cen­ tral subject of this paintings. notwithstanding, a few of the notions of integration built listed below are in detail attached with a selected application-the illustration of evolutions by means of func­ tional integrals. The illustration of a perturbation to the warmth semigroup when it comes to Wiener degree is named the Feynman-Kac formulation, however the time period has a much broader which means within the current paintings. ordinarily, such representations were used to acquire analytic information regarding perturbations to unfastened evolutions in its place to arguments with a extra operator-theoretic flavour. No functions of this kind are given the following. it truly is an un­ derlying assumption of the presentation of this fabric that representations of the character of the Feynman-Kac formulation are worthy acquiring, and within the strategy of acquiring them, we could be resulted in new, in all probability fertile mathematical structures-a view principally prompted by means of the pervasive use of course integrals in quantum physics.

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X tk E Bk,n} with n = 1, ... , m and m = 1,2, ... , of pairwise disjoint basic events with < tl < ... 8) L Mlk (Xtl n=1 E BI,n, ... n)X E Kx,tk' 36 2. o and m MIl (XII E BI,n)x = L L n=1 Q(BI,i)S(II)X E KX,lI' n=1 Now suppose that the assertion is true for k = j. We show that it is true for k = j + 1. Let m be a positive integer, and suppose that {XII E BI,n, ... , X lj +1 E Bj+I,n}, n = 1, ... , m is a family of pairwise disjoint basic events with 0 < II < ... < tj+1 :s t. By making a suitable decomposition of each event, and appealing to the additivity of M lj + l , it may be assumed from the outset that the sets Bj+I,n are identical for np :s n < np+l' p = 1, ...

Of subsets of :E x :E; any set in [E{O,tl] may be expresses as such a pairwise disjoint union. In the notation mentioned earlier, for a bounded E measurable function g, we write Q (g) for the definite integral g d Q of g with respect to the spectral measure Q. In the case that Q is multiplication by characteristic functions, Q(g) is multiplication by g. Jr. 1 Lemma. 11) IL k Q(gj)S(t)Q(jj)II£(E) j=1 :s cll L jj ® gj 1100' j=1 holdsforallboundedC-valuedE-measurablefunctions jj, gj, j = 1, ... , k, defined on :E and all k = 1, 2, ....

The operator valued measure M tE is the tensor product M t ® I dE of M t with the identity operator I dE acting on E. The evolution process M; = (n, (Ft}t~o, (MtE}t~O; (Xt}t~o) with state space LP(r, E, fL; E) may be viewed as the tensor product of the evolution process Mp with IdE. 7 Example. (Wiener process) Let n be the collection of all continuous functions = w(s) for every WEn. The aalgebra generated by X s , 0 S sSt, is denoted by St. Let S be the Borel a-algebra of n under the topology of uniform convergence on compact subsets of [0, (0).

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