Stochastic dilations of uniformly continuous completely positive semigroups

Hudson , R. L. ; Parthasarathy, K. R. (1984) Stochastic dilations of uniformly continuous completely positive semigroups Acta Applicandae Mathematica, 2 (3-4). pp. 353-378. ISSN 0167-8019

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Official URL: http://www.springerlink.com/content/pgl761334556qq...

Related URL: http://dx.doi.org/10.1007/BF02280859

Abstract

For an arbitrary uniformly continuous completely positive semigroup (F t:t≥0) on the space B(h0) of bounded operators on a Hilbert space h0, we construct a family (U(t):t≥0) of unitary operators on a Hilbert space b0 = h0 ⊗ b and a conditional expectation E0 from B(b0) to B(h0) , such that, for arbitrary t≥0, X ∊ B (h0))Ft(X) = E0[U(t)X ⊗IU(t)†] . The unitary operators U(t) satisfy a stochastic differential equation involving a non commutative generalization of infinite dimensional Brownian motion. They do not form a semi group.

Item Type:Article
Source:Copyright of this article belongs to Springer.
Keywords:Completely Positive Semi group; Operators on Hilbert Space; Conditional Expectation; Stochastic Differential Equation; ∞-Dimensional Brownian Motion; Fock Space; Itô Product Formula; Stochastic Dilation
ID Code:36208
Deposited On:25 May 2011 13:34
Last Modified:25 May 2011 13:34

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