Solving quantum stochastic differential equations with unbounded coefficients

We demonstrate a method for obtaining strong solutions to the right Hudson–Parthasarathy quantum stochastic differential equationdUt=FβαUt dΛαβ(t), U0=1where 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 Dom C1/2 is left invariant by the operators Ut, 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.