  Statistical properties of Brownian motion that arise by analyzing,
separately, trajectories over which the system energy increases (upside) or
decreases (downside) with respect to a threshold energy level, are derived.
This selective analysis is applied to examine transport properties of a
nonequilibrium Brownian process that is coupled to multiple thermal sources
characterized by different temperatures. Distributions, moments, and
correlation functions of a free particle that occur during upside and downside
events are investigated for energy activation and energy relaxation processes,
and also for positive and negative energy fluctuations from the average energy.
The presented results are sufficiently general and can be applied without
modification to standard Brownian motion. This article focuses on the
mathematical basis of this selective analysis. In subsequent articles in this
series we apply this general formalism to processes in which heat transfer
between thermal reservoirs is mediated by activated rate processes that take
place in a system bridging them.
