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The fluctuation-dissipation theorem, in the Kubo original formulation, isbased on the decomposition of the thermal agitation forces into a dissipativecontribution and a stochastically fluctuating term. This decomposition can beavoided by introducing a stochastic velocity field, with correlation propertiesderiving from linear response theory. Here, we adopt this field as thecomprehensive hydrodynamic/fluctuational driver of the kinematic equations ofmotion. With this description, we show that the Langevin equations for aBrownian particle interacting with a solvent fluid become particularly simpleand can be applied even in those cases in which the classical approach, basedon the concept of a stochastic thermal force, displays intrinsic difficultiese.g., in the presence of the Basset force. We show that a convenient way fordescribing hydrodynamic/thermal fluctuations is by expressing them in the formof Extended Poisson-Kac Processes possessing prescribed correlation propertiesand a continuous velocity density function. We further highlight the importanceof higher-order correlation functions in the description of the stochastichydrodynamic velocity field with special reference to short-time properties ofBrownian motion. We conclude by outlining some practical implications inconnection with the statistical description of particle motion in confinedgeometries.

Stochastic hydrodynamic velocity field and the representation of Langevin equations