We establish a formula for the distribution of loop-erased random walks at certain random times. Several classical results on spanning trees, including Wilson's algorithm, follow easily, as well as a method to construct random Hamiltonian cycles.

We introduce the concept of a mild solution for the right Hudson-Parthasarathy quantum stochastic differential equation, prove existence and uniqueness results, and show the correspondence between our definition and similar ideas in the theory of classical stochastic differential equations. The…