Brownian motion apart once the particle is in motion. Also, there would be a distribution of different possible Δ"V"s instead of always just one in a realistic situation. The diffusion equation yields an approximation of the time evolution of the probability density function associated to the position of the particle going under a Brownian movement under the physical definition. The approximation is valid on short timescales. The time evolution of the position of the Brownian particle itself is best described using Langevin equation, an equation which involves a random force field representing the effect of the thermal fluctuations of the solvent on
Brownian excursion Groeneboom (1983) and Pitman (1983) give decompositions of Brownian motion formula_5 in terms of i.i.d Brownian excursions and the least concave majorant (or greatest convex minorant) of formula_5. For an introduction to Itô's general theory of Brownian excursions and the Itô Poisson process of excursions, see Revuz and Yor (1994), chapter XII. The Brownian excursion area arises in connection with the enumeration of connected graphs, many other problems in combinatorial theory; see e.g.,, and the limit distribution of the Betti numbers of certain varieties in cohomology theory. Takacs (1991a) shows that formula_19 has density where formula_21 are the zeros of