A note on Harris' ergodic theorem, controllability and perturbations of harmonic networks

Abstract: We show that elements of control theory, together with an application of Harris' ergodic theorem, provide an alternate method for showing exponential convergence to a unique stationary measure for certain classes of networks of quasi-harmonic classical oscillators coupled to heat baths. With the system of oscillators expressed in the form $\mathrm{d} X_t = A X_t \,\mathrm{d} t + F(X_t) \,\mathrm{d} t + B \,\mathrm{d} W_t$ in $\mathbf{R}^d$, where $A$ encodes the harmonic part of the force and $-F$ corresponds to the gradient of the anharmonic part of the potential, the hypotheses under which we obtain exponential mixing are the following: $A$ is dissipative, the pair $(A,B)$ satisfies the Kalman condition, $F$ grows sufficiently slowly at infinity (depending on the dimension $d$), and the vector fields in the equation of motion satisfy the weak H\"ormander condition in at least one point of the phase space.