Solving quantum stochastic differential equations with unbounded coefficients

We demonstrate a method for obtaining strong solutions to the right Hudson-Parthasarathy quantum stochastic differential equation dU(t) = (FbetaU1)-U-alpha dLambda(alpha)(beta)(t), U-0 = 1 where U is a contraction operator process, and the matrix of coefficients [F-beta(alpha)] consists of unbounded operators. This is achieved whenever there is a positive self-adjoint reference operator C that behaves well with respect to the F-beta(alpha), allowing us to prove that Dom C-1/2 is left invariant by the operators U,, 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. (C) 2002 Elsevier Science (USA). All rights reserved.