*J. Klafter and I. M. Sokolov*

- Published in print:
- 2011
- Published Online:
- December 2013
- ISBN:
- 9780199234868
- eISBN:
- 9780191775024
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199234868.001.0001
- Subject:
- Physics, Soft Matter / Biological Physics

The name “random walk” for a problem of a displacement of a point in a sequence of independent random steps was coined by Karl Pearson in 1905 in a question posed to readers of “Nature”. The same ...
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The name “random walk” for a problem of a displacement of a point in a sequence of independent random steps was coined by Karl Pearson in 1905 in a question posed to readers of “Nature”. The same year, a similar problem was formulated by Albert Einstein in one of his Annus Mirabilis works. Even earlier problem was posed by Louis Bachelier in his thesis devoted to the theory of financial speculations in 1900. Nowadays theory of random walks was proved useful in physics and chemistry (diffusion, reactions, mixing in flows), economics, biology (from animal spread to motion of subcellular structures) and in many other disciplines. The random walk approach serves not only as a model of simple diffusion but of many complex sub‐ and superdiffusive transport processes as well. This book discusses main variants of the random walks and gives the most important mathematical tools for their theoretical description.Less

The name “random walk” for a problem of a displacement of a point in a sequence of independent random steps was coined by Karl Pearson in 1905 in a question posed to readers of “Nature”. The same year, a similar problem was formulated by Albert Einstein in one of his *Annus Mirabilis* works. Even earlier problem was posed by Louis Bachelier in his thesis devoted to the theory of financial speculations in 1900. Nowadays theory of random walks was proved useful in physics and chemistry (diffusion, reactions, mixing in flows), economics, biology (from animal spread to motion of subcellular structures) and in many other disciplines. The random walk approach serves not only as a model of simple diffusion but of many complex sub‐ and superdiffusive transport processes as well. This book discusses main variants of the random walks and gives the most important mathematical tools for their theoretical description.

*J. Klafter and I.M. Sokolov*

- Published in print:
- 2011
- Published Online:
- December 2013
- ISBN:
- 9780199234868
- eISBN:
- 9780191775024
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199234868.003.0003
- Subject:
- Physics, Soft Matter / Biological Physics

Up to now the book has considered the displacement and the return properties of a walker as functions of the number of steps. In physics and chemistry we however are mostly interested in the behavior ...
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Up to now the book has considered the displacement and the return properties of a walker as functions of the number of steps. In physics and chemistry we however are mostly interested in the behavior of the corresponding properties as functions of time. Translating steps into time and back is done by assuming that a walker waits for the next step for a time t distributed according to a known density. Such a process is known as continuous time random walk (CTRW). The CTRW formalism is used to calculate the distribution of displacements, as well as first passage and return times. Special attention is paid to the case when the mean waiting time diverges, as a one often used in applications.Less

Up to now the book has considered the displacement and the return properties of a walker as functions of the number of steps. In physics and chemistry we however are mostly interested in the behavior of the corresponding properties as functions of time. Translating steps into time and back is done by assuming that a walker waits for the next step for a time t distributed according to a known density. Such a process is known as continuous time random walk (CTRW). The CTRW formalism is used to calculate the distribution of displacements, as well as first passage and return times. Special attention is paid to the case when the mean waiting time diverges, as a one often used in applications.

*Luis Caffarelli and Luis Silvestre*

*Charles Fefferman, Alexandru D. Ionescu, D. H. Phong, Stephen Wainger, Charles Fefferman, Alexandru D. Ionescu, D. H. Phong, and Stephen Wainger (eds)*

- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691159416
- eISBN:
- 9781400848935
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691159416.003.0004
- Subject:
- Mathematics, Numerical Analysis

This chapter studies evolution problems that are related to continuous time random walks (CTRW), having a discontinuous path for which both the jumps and the time elapsed in between them are random. ...
More

This chapter studies evolution problems that are related to continuous time random walks (CTRW), having a discontinuous path for which both the jumps and the time elapsed in between them are random. These processes are governed by a generalized master equation which is nonlocal both in space and time. To illustrate, the chapter considers kernels K(t, x, s, y) in a particular function. Here, studying correlated kernels provides a more flexible framework where more interesting physical phenomena can be observed, and more subtle mathematical questions appear. The regularity estimates are in fact more interesting (harder mathematically) when the jumps in space and the waiting times are strongly correlated.Less

This chapter studies evolution problems that are related to continuous time random walks (CTRW), having a discontinuous path for which both the jumps and the time elapsed in between them are random. These processes are governed by a generalized master equation which is nonlocal both in space and time. To illustrate, the chapter considers kernels *K*(*t*, *x*, *s*, *y*) in a particular function. Here, studying correlated kernels provides a more flexible framework where more interesting physical phenomena can be observed, and more subtle mathematical questions appear. The regularity estimates are in fact more interesting (harder mathematically) when the jumps in space and the waiting times are strongly correlated.