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 = F _β ^α U _t dΛ _α ^β (t), U ₀ = 1 where U is a contraction operator process, and the matrix of coefficients [F _β ^α ] 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 _β ^α , allowing us to prove that Dom C ^1/2 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.