  We consider a quantum Brownian particle interacting with two harmonic baths,
which is then perturbed by a cubic coupling linking the particle and the baths.
This cubic coupling induces non-linear dissipation and noise terms in the
influence functional/master equation of the particle. Its effect on the
Out-of-Time-Ordered Correlators (OTOCs) of the particle cannot be captured by
the conventional Feynman-Vernon formalism.We derive the generalised influence
functional which correctly encodes the physics of OTO fluctuations, response,
dissipation and decoherence. We examine an example where Markovian
approximation is valid for the OTO dynamics. If the original cubic coupling has
a definite time-reversal parity, the leading order OTO influence functional is
completely determined by the couplings in the usual master equation via OTO
generalisation of Onsager-Casimir relations. New OTO fluctuation-dissipation
relations connect the non-Gaussianity of the thermal noise to the thermal
jitter in the damping constant of the Brownian particle.
