It isn't right-in fact the statement is meaningless. There isn't any such notion of ``velocity phase space'' or ``position phase space''-there's, just, phase space, that's parametrized by coordinates and momenta-and which is which doesn't matter. However phase space isn't a relativistically invariant notion, so care must be taken in constructing covariant quantities, that can be assigned physical meaning.
This has been studied within the context of stochastic quantization of field theory. Relativistic Browinian motion is just a historical term that refers to the free scalar field, or the free relativistic particle, in this context.
Phase space is in momentum (or rigidity) and spatial coordinates. Particles move, and thus are subject to the Special Theory. In Boltzmann's equation, momenta are usually converted to energy since cross sections are expressed in energy, and decay constants (e.g.pions or muons) in terms of the Lorenz factor.
Space time is a substance created as part of the Big Bang. The presence of this substance explains the presence of energy in free space.
Inherent in this substance are the characeteristic electrical constants of the permability of free space and the permittivity of free space which together make the speed of light.
So that is why you need velocity space
Se my papers in reasearch Gate
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