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Deuterium burning (or deuterium fusion) is the fusion of deuterium nuclei. Some the proton-proton chains include this as a step, and a body containing some deuterium can trigger deuterium fusion at a lower temperature than that necessary to start 1H fusion. Thus deuterium burning begins sooner, and can occur even if hydrogen burning never occurs in the body.
The defining line between giant planet and brown dwarf is that the latter is sufficiently massive that the temperature rises to the point that deuterium burning is triggered. This requires deuterium in the body, generally true for protostars, but such bodies have only sufficient deuterium to sustain fusion for a relatively brief time period. If the temperature grows hot enough that hydrogen burning begins, the body is defined to be a main sequence star rather than a brown dwarf.