Brownian motion but rather a singular behavior in which the particle doesn't move at all. The Brownian motion can be modeled by a random walk. Random walks in porous media or fractals are anomalous. In the general case, Brownian motion is a non-Markov random process and described by stochastic integral equations. The French mathematician Paul Lévy proved the following theorem, which gives a necessary and sufficient condition for a continuous R-valued stochastic process "X" to actually be "n"-dimensional Brownian motion. Hence, Lévy's condition can actually be used as an alternative definition of Brownian motion. Let "X" = ("X", ..., "X") be a
Brownian motion motion can be measured as "v" = Δ"x"/Δ"t", when Δ"t" « "τ", where "τ" is the momentum relaxation time. In 2010, the instantaneous velocity of a Brownian particle (a glass microsphere trapped in air with optical tweezers) was measured successfully. The velocity data verified the Maxwell–Boltzmann velocity distribution, and the equipartition theorem for a Brownian particle. In stellar dynamics, a massive body (star, black hole, etc.) can experience Brownian motion as it responds to gravitational forces from surrounding stars. The rms velocity "V" of the massive object, of mass "M", is related to the rms velocity formula_34 of the background