  The dynamical correlations of nonlocal operators in general quadratic open
fermion systems is still a challenging problem. Here we tackle this problem by
developing a new formulation of open fermion many-body systems, namely, the
characteristic function approach. Illustrating the technique, we analyze a
finite Kitaev chain with boundary dissipation and consider anyon-type nonlocal
excitations. We give explicit formula for the Green's functions, demonstrating
an asymmetric light cone induced by the anyon statistical parameter and an
increasing relaxation rate with this parameter. We also analyze some other
types of nonlocal operator correlations such as the full counting statistics of
the charge number and the Loschmidt echo in a quench from the vacuum state. The
former shows clear signature of a nonequilibrium quantum phase transition,
while the later exhibits cusps at some critical times and hence demonstrates
dynamical quantum phase transitions.
