Journal of Fourier analysis and applications 11 2 , , Stochastic processes and their applications 69 1 , , Journal of Theoretical Probability 15 2 , , Israel journal of mathematics 68 3 , , Probability theory and related fields 71 3 , , Random walks on infinite graphs and groups W Woess Cambridge university press , Random walks on infinite graphs and groups—a survey on selected topics W Woess Bulletin of the London Mathematical Society 26 1 , , Denumerable Markov chains: generating functions, boundary theory, random walks on trees W Woess European Mathematical Society , Amenability, unimodularity, and the spectral radius of random walks on infinite graphs PM Soardi, W Woess Mathematische Zeitschrift 1 , , A Wiener process is the scaling limit of random walk in dimension 1.
This means that if you take a random walk with very small steps, you get an approximation to a Wiener process and, less accurately, to Brownian motion. As the step size tends to 0 and the number of steps increases proportionally , random walk converges to a Wiener process in an appropriate sense. Formally, if B is the space of all paths of length L with the maximum topology, and if M is the space of measure over B with the norm topology, then the convergence is in the space M. Similarly, a Wiener process in several dimensions is the scaling limit of random walk in the same number of dimensions.
A random walk is a discrete fractal a function with integer dimensions; 1, 2, 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. A Wiener process enjoys many symmetries random walk does not. For example, a Wiener process walk is invariant to rotations, but the random walk is not, since the underlying grid is not random walk is invariant to rotations by 90 degrees, but Wiener processes are invariant to rotations by, for example, 17 degrees too.
This means that in many cases, problems on a random walk are easier to solve by translating them to a Wiener process, solving the problem there, and then translating back.
- Random walk - Wikipedia.
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On the other hand, some problems are easier to solve with random walks due to its discrete nature. Random walk and Wiener process can be coupled , namely manifested on the same probability space in a dependent way that forces them to be quite close. The convergence of a random walk toward the Wiener process is controlled by the central limit theorem , and by Donsker's theorem. This corresponds to the Green's function of the diffusion equation that controls the Wiener process, which suggests that, after a large number of steps, the random walk converges toward a Wiener process.
In 3D, the variance corresponding to the Green's function of the diffusion equation is:. By equalizing this quantity with the variance associated to the position of the random walker, one obtains the equivalent diffusion coefficient to be considered for the asymptotic Wiener process toward which the random walk converges after a large number of steps:.
A random walk having a step size that varies according to a normal distribution is used as a model for real-world time series data such as financial markets. The Black—Scholes formula for modeling option prices, for example, uses a Gaussian random walk as an underlying assumption. For steps distributed according to any distribution with zero mean and a finite variance not necessarily just a normal distribution , the root mean square translation distance after n steps is. But for the Gaussian random walk, this is just the standard deviation of the translation distance's distribution after n steps.
Some diffusions in random environment are even proportional to a power of the logarithm of the time, see for example Sinai's walk or Brox diffusion. It is also related to the vibrational density of states,   diffusion reactions processes  and spread of populations in ecology. The information rate of a Gaussian random walk with respect to the squared error distance, i. As mentioned the range of natural phenomena which have been subject to attempts at description by some flavour of random walks is considerable, in particular in physics   and chemistry,  materials science ,   biology  and various other fields.
In all these cases [ which? A number of types of stochastic processes have been considered that are similar to the pure random walks but where the simple structure is allowed to be more generalized.
glucinamchicti.cfility - A random walk on an infinite graph is recurrent iff ? - MathOverflow
The pure structure can be characterized by the steps being defined by independent and identically distributed random variables. Building on the analogy from the earlier section on higher dimensions, assume now that our city is no longer a perfect square grid. When our person reaches a certain junction, he picks between the variously available roads with equal probability.
Thus, if the junction has seven exits the person will go to each one with probability one-seventh.
This is a random walk on a graph. Will our person reach his home? It turns out that under rather mild conditions, the answer is still yes. For example, if the lengths of all the blocks are between a and b where a and b are any two finite positive numbers , then the person will, almost surely, reach his home. Notice that we do not assume that the graph is planar , i. One way to prove this result is using the connection to electrical networks.
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Take a map of the city and place a one ohm resistor on every block. Now measure the "resistance between a point and infinity. This is now a finite electrical network, and we may measure the resistance from our point to the wired points. Take R to infinity. The limit is called the resistance between a point and infinity. It turns out that the following is true an elementary proof can be found in the book by Doyle and Snell :.
Theorem : a graph is transient if and only if the resistance between a point and infinity is finite. It is not important which point is chosen if the graph is connected. In other words, in a transient system, one only needs to overcome a finite resistance to get to infinity from any point. In a recurrent system, the resistance from any point to infinity is infinite.
This characterization of transience and recurrence is very useful, and specifically it allows us to analyze the case of a city drawn in the plane with the distances bounded. A random walk on a graph is a very special case of a Markov chain. Unlike a general Markov chain, random walk on a graph enjoys a property called time symmetry or reversibility. Roughly speaking, this property, also called the principle of detailed balance , means that the probabilities to traverse a given path in one direction or the other have a very simple connection between them if the graph is regular , they are just equal.
This property has important consequences. Starting in the s, much research has gone into connecting properties of the graph to random walks. A significant portion of this research was focused on Cayley graphs of finitely generated groups. In many cases these discrete results carry over to, or are derived from manifolds and Lie groups. These include the distribution of first  and last hitting times  of the walker, where the first hitting time is given by the first time the walker steps into a previously visited site of the graph, and the last hitting time corresponds the first time the walker cannot perform an additional move without revisiting a previously visited site.
A good reference for random walk on graphs is the online book by Aldous and Fill. For groups see the book of Woess. See the book of Hughes, the book of Revesz, or the lecture notes of Zeitouni. We can think about choosing every possible edge with the same probability as maximizing uncertainty entropy locally.
Recommend Documents. Random Walks and Diffusions on Graphs and Databases: An Introduction Springer Complexity Springer Complexity is an interdisciplinary program publishing the best research and academic-level Random walks and turbulence Probability on graphs: Random processes on graphs and lattices This page intentionally left blank Probability on Graphs This introduction to some of the principal models in the the Random walks and electric networks Random walks and electric networks arXiv:math.