Solving quantum stochastic differential equations with unbounded coefficients
The authors demonstrate a method for obtaining strong solutions to the right Hudson-Parthasarathy quantum stochastic differential equation \(dU_t = F^\alpha_\beta U_td\Lambda^\beta_\alpha(t), \;U_0=1\) where \(U\) is a contraction operator process, \(\Lambda^\beta_\alpha\) are the fundamental noise processes, and the coefficients \(F^\alpha_\beta\) are unbounded operators. This is achieved whenever there is a positive self-adjoint reference operator \(C\) that behaves well with respect to the \(F^\alpha_\beta\) allowing them to prove that the domain of \((C^{1/2})\) is left invariant by the operators \(U_t\), thereby giving rigorous meaning to the formal expression above. The germ of the idea is as follows: assuming that the associated left quantum stochastic differential equation \[ dV_t = V_t(F^\alpha_\beta)^*d\Lambda^\alpha_\beta(t), \;V_0=1 \] can be solved, and also that there is a self-adjoint operator \(C\) such that, letting \(C_\varepsilon=C(1+\varepsilon C)^2\), the operators \((F^\alpha_\beta)^*C_\varepsilon^{1/2}\) are bounded, they show that the process \(C_\varepsilon^{1/2} U\) satisfies the right quantum stochastic differential equation \[ dC_\varepsilon^{1/2} U_t = [(F^\alpha_\beta)^* C_\varepsilon^{1/2}]^* U_t d\Lambda^\beta_\alpha(t),\qquad C_\varepsilon^{1/2} U_0 = C_\varepsilon^{1/2}. \] Then, if the ``reference operator'' \(C\) satisfies a form inequality that can be written heuristically as \( \theta(C) \leq b C\), they obtain a priori estimates on the continuity of each \(U_t\) with respect to the graph norm of \(C^{1/2}\) and hence obtain information about the range of \(U_t\). Conditions under which the solution \(U\) is an isometry or coisometry process are also discussed. The above results are applied to construct unital \({}^*\)-homomorphic dilations of (quantum) Markov semigroups arising in probability and physics.
scientific article; zbMATH DE number 1901771
scientific article