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A random walk is a discrete fractal (a function with integer dimensions; 1, 2, ...), but a Wiener process trajectory is a true fractal, and there is a connection between the two. For example, take a random walk until it hits a circle of radius r times the step length. The average number of steps it performs is r 2.
In mathematics, a continuous-time random walk (CTRW) is a generalization of a random walk where the wandering particle waits for a random time between jumps. It is a stochastic jump process with arbitrary distributions of jump lengths and waiting times.
For random walks on a hyperbolic group, under the finite entropy assumption on the step distribution which always hold for a simple walk (a more general condition is that the first moment be finite) the Poisson boundary is always equal to the Gromov boundary when equipped with the hitting probability measure.
The random walk model of consumption was introduced by economist Robert Hall. [1] This model uses the Euler numerical method to model consumption. He created his consumption theory in response to the Lucas critique. Using Euler equations to model the random walk of consumption has become the dominant approach to modeling consumption. [2]
An unbiased random walk, in any number of dimensions, is an example of a martingale. For example, consider a 1-dimensional random walk where at each time step a move to the right or left is equally likely. A gambler's fortune (capital) is a martingale if all the betting games which the gambler plays are fair.
The actual random walk obeys a stochastic equation of motion, but its probability density function (PDF) obeys a deterministic equation. PDFs of random walks can be formulated in terms of the (discrete in space) master equation [1] [12] [13] and the generalized master equation [3] or the (continuous in space and time) Fokker Planck equation [37] and its generalizations. [10]
Maximal entropy random walk (MERW) is a popular type of biased random walk on a graph, in which transition probabilities are chosen accordingly to the principle of maximum entropy, which says that the probability distribution which best represents the current state of knowledge is the one with largest entropy.
A loop-erased random walk in 2D for steps. In mathematics, loop-erased random walk is a model for a random simple path with important applications in combinatorics, physics and quantum field theory. It is intimately connected to the uniform spanning tree, a model for a random tree.