Brownian motion diverges as the window shrinks, thus rendering the calculation a singular perturbation problem. Brownian motion Brownian motion or pedesis (from "leaping") is the random motion of particles suspended in a fluid (a liquid or a gas) resulting from their collision with the fast-moving molecules in the fluid. This pattern of motion typically alternates random fluctuations in a particle's position inside a fluid sub-domain with a relocation to another sub-domain. Each relocation is followed by more fluctuations within the new closed volume. This pattern describes a fluid at thermal equilibrium, defined by a given temperature. Within such fluid there exists no
Fractional Brownian motion Suppose we want to simulate the values of the "fBM" at times formula_12 using the Cholesky decomposition method. In order to compute formula_15, we can use for instance the Cholesky decomposition method. An alternative method uses the eigenvalues of formula_16: Note that the result is real-valued because formula_36. Note that since the eigenvectors are linearly independent, the matrix formula_39 is invertible. It is also known that where "B" is a standard Brownian motion and Where formula_47 is the Euler hypergeometric integral. Say we want to simulate an "fBm" at points formula_48. The integral may be efficiently computed by Gaussian quadrature.