Identities for nonlinear memory kernels

Perturbing a system far away from equilibrium via a time-dependent protocol can formally be described by a nonlinear Volterra series expansion. Here we derive identities for the nonlinear memory kernels arising in such nonlinear expansion, including the possibility of a nonlinear coupling between perturbation and system. These identities rely on local detailed balance, and they include the fluctuation dissipation theorem as the lowest-order identity. We test them in simulations for driven over- and underdamped Brownian particles. These identities for memory kernels can be recast in a series relation for the nonequilibrium cumulants of the observable conjugate to the driving and the observable described by the Volterra series. We discuss the relation to previously found identities.