Peer-Reviewed Journal Details
Mandatory Fields
Fagnola, Franco; Wills, Stephen J. ;
Journal of Functional Analysis
Solving quantum stochastic differential equations with unbounded coefficients
Optional Fields
Quantum stochastic; Stochastic differential equation; Stochastic cocycle; Birth and death process; Inverse oscillator; Diffusion process
We demonstrate a method for obtaining strong solutions to the right Hudson–Parthasarathy quantum stochastic differential equation dU = FU dLambda where U is a contraction operator process, and the matrix of coefficients [Fβα] consists of unbounded operators. This is achieved whenever there is a positive self-adjoint reference operator C that behaves well with respect to the Fβα, allowing us to prove that the domain of the square root of C is left invariant by the operators U_t, thereby giving rigorous meaning to the formal expression above. We give conditions under which the solution U is an isometry or coisometry process, and apply these results to construct unital *-homomorphic dilations of (quantum) Markov semigroups arising in probability and physics.
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