  To integrate hydrodynamic fluctuations, namely thermal fluctuations of
hydrodynamics, into dynamical models of high-energy nuclear collisions based on
relativistic hydrodynamics, the property of the hydrodynamic fluctuations given
by the fluctuation-dissipation relation should be carefully investigated. The
fluctuation-dissipation relation for causal dissipative hydrodynamics with the
finite relaxation time is naturally given in the integral form of the
constitutive equation by the linear-response theory. While, the differential
form of the constitutive equation is commonly used in analytic investigations
and dynamical calculations for practical reasons. We give the
fluctuation-dissipation relation for the general linear-response differential
form and discuss the restrictions to the structure of the differential form,
which comes from the causality and the positive semi-definiteness of the noise
autocorrelation, and also the relation of those restrictions to the cutoff
scale of the hydrodynamic fluctuations. We also give the
fluctuation-dissipation relation for the integral form in non-static and
inhomogeneous background by introducing new tensors, the pathline projectors.
We find new modification terms to the fluctuation-dissipation relation for the
differential form in non-static and inhomogeneous background which are
particularly important in dynamical models to describe rapidly expanding
systems.
