Stochastic Dilations of Uniformly Continuous Completely Positive Semigroups

R. L. Hudson and K. R. Parthasarathy
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R. L. Hudson: University of Nottingham, Mathematics Department
K. R. Parthasarathy: Indian Statistical Institute

A chapter in Positive Semigroups of Operators, and Applications, 1984, pp 353-378 from Springer

Abstract: Abstract For an arbitrary uniformly continuous completely positive semigroup (ℑ t : t≥ 0) on the space B(ɧ0) of bounded operators on a Hilbert space ɧ0, we construct a family (U(t): t ≥ 0) of unitary operators on a Hilbert space ℌ0 = ɧ0 ⊗ ℌ and a conditional expectation E0 from B(ℌ0) to B(ℌ0), such that, for arbitrary t ≥0, X ∈ B(ɧ0) ℑ t (X) = E0[U(t)X ⊗ IU(t)†]. The unitary operators U(t) satisfy a stochastic differential equation involving a noncommutative generalisation of infinite dimensional Brownian motion. They do not form a semigroup.

Keywords: Completely positive semigroup; operators on Hilbert space; conditional expectation; stochastic differential equation; ∞-dimensional Brownian motion; Fock space; Itô product formula; stochastic dilation (search for similar items in EconPapers)
Date: 1984
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DOI: 10.1007/978-94-009-6484-6_6

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