  When a probe particle immersed in a fluid with nonlinear interactions is
subject to strong driving, the cumulants of the stochastic force acting on the
probe are nonlinear functionals of the driving protocol. We present a Volterra
series for these nonlinear functionals, by applying nonlinear response theory
in a path integral formalism, where the emerging kernels are shown to be
expressed in terms of connected equilibrium correlation functions. The first
cumulant is the mean force, the second cumulant characterizes the
non-equilibrium force fluctuations (noise), and higher order cumulants quantify
non-Gaussian fluctuations. We discuss the interpretation of this formalism in
relation to Langevin dynamics. We highlight two example scenarios of this
formalism: i) For a particle driven with prescribed trajectory, the formalism
yields the non-equilibrium statistics of the interaction force with the fluid.
ii) For a particle confined in a moving trapping potential, the formalism
yields the non-equilibrium statistics of the trapping force. In simulations of
a model of nonlinearly interacting Brownian particles, we find that nonlinear
phenomena, such as shear-thinning and oscillating noise covariance, appear in
third or second order response, respectively.
