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 surface Brownian motion, Brownian surfaces are named after 19th-century biologist Robert Brown. Efficient generation of fractional Brownian surfaces poses significant challenges. Since the Brownian surface represents a Gaussian process with a nonstationary covariance function, one can use the Cholesky decomposition method. A more efficient method is Stein's method, which generates an auxiliary stationary Gaussian process using the circulant embedding approach and then adjusts this auxiliary process to obtain the desired nonstationary Gaussian process. The figure below shows three typical realizations of fractional Brownian surfaces for different values of the roughness or Hurst parameter. The Hurst parameter is always between zero and