For a free Brownian particle in contact with multiple reservoirs, each bath contributes a fraction gamma_k T_k / (gamma T) of the particle's energy change during both energy-gaining (upside) and energy-losing (downside) events.
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Upside/Downside statistical mechanics of nonequilibrium Brownian motion. II. Heat transfer and energy partitioning of a free particle
For a free Brownian particle in contact with multiple reservoirs, each bath contributes a fraction gamma_k T_k / (gamma T) of the particle's energy change during both energy-gaining (upside) and energy-losing (downside) events.