Fermion Ito's formula and stochastic evolutions

Abstract

An Ito product formula is proved for stochastic integrals against Fermion Brownian motion, and used to construct unitary processes satisfying stochastic differential equations. As in the corresponding Boson theory [10,11] these give rise to stochastic dilations of completely positive semigroups.

Fermion Ito's Formula II: The Gauge Process in Fermion Fock Space

“…Preliminaries (cf. [2], [9]) A Hilbert space 3? is said to be Z 2 -graded if it may be written J^ = ^f+@^^ where J^+ and Jf _ are called the even and odd subspaces (respectively).…”

“…The stochastic calculus constructed in [2] for fermion Brownian motion is augmented through the inclusion of stochastic integration with respect to the gauge process. The solutions of certain non-commutative stochastic differential equations are used to construct dilations of contraction semigroups on a Hilbert space f)o and of uniformly continuous, completely positive semigroups on -ff(Ijo).…”

“…In the recent paper [2], the present author together with R 0 L 0 Hudson developed a stochastic calculus on fermion Fock space over L 2 (R + ) in which the Brownian motion process of classical stochastic calculus was replaced by fermion Brownian motion of variance 1 [4] i.e. the process formed from pairs (A t , AD of annihilation and creation operators smeared by the indicator function of the interval [0, t) which satisfy the anticommutation relation With regard to its sister theory previously developed in [10] by R. L. Hudson and K. R. Parthasarathy for boson Fock space, the scheme of [2] contained two major defects ( i ) The structure of the algebra of stochastic integrals involved unnatural parity assumptions.…”

“…the process formed from pairs (A t , AD of annihilation and creation operators smeared by the indicator function of the interval [0, t) which satisfy the anticommutation relation With regard to its sister theory previously developed in [10] by R. L. Hudson and K. R. Parthasarathy for boson Fock space, the scheme of [2] contained two major defects ( i ) The structure of the algebra of stochastic integrals involved unnatural parity assumptions.…”

“…The organization of the present paper is as follows; after collecting together some useful algebraic facts in § 1 we proceed to define the fermion gauge operator in § 2 and examine its relevant properties. As in [2] our strategy is to use /2-particle vectors in place of exponential vectors to define our operators. In § 4 and § 5 we develop our stochastic calculus and find that the extended version of (0.…”