### Abstract

We study a random walk with nearest neighbor transitions on a one-dimensional lattice. The walk starts at the origin, as does a dividing line which moves with constant speed gamma , but the outward transition probabilities p//A and p//B differ on the right- and left-hand sides of the dividing line. This problem is solved formally by taking advantage of the analytical properties in the complex plane of an added variable generating function, and it is found that p//A, p//B space decomposes into four regions of distinct qualitative properties. The asymptotic probability of the walk being to the right of the moving boundary is obtained explicitly in three of the four regions. However, analysis in the fourth region is a sensitive function of the denominator of the rational fraction gamma , and so we conclude with a number of special cases which can be solved in closed form.

Original language | English (US) |
---|---|

Pages (from-to) | 822-830 |

Number of pages | 9 |

Journal | SIAM Journal on Applied Mathematics |

Volume | 47 |

Issue number | 4 |

State | Published - Aug 1987 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*SIAM Journal on Applied Mathematics*,

*47*(4), 822-830.

**PIECEWISE HOMOGENEOUS RANDOM WALK WIH A MOVING BOUNDARY.** / Percus, Ora E.; Percus, Jerome.

Research output: Contribution to journal › Article

*SIAM Journal on Applied Mathematics*, vol. 47, no. 4, pp. 822-830.

}

TY - JOUR

T1 - PIECEWISE HOMOGENEOUS RANDOM WALK WIH A MOVING BOUNDARY.

AU - Percus, Ora E.

AU - Percus, Jerome

PY - 1987/8

Y1 - 1987/8

N2 - We study a random walk with nearest neighbor transitions on a one-dimensional lattice. The walk starts at the origin, as does a dividing line which moves with constant speed gamma , but the outward transition probabilities p//A and p//B differ on the right- and left-hand sides of the dividing line. This problem is solved formally by taking advantage of the analytical properties in the complex plane of an added variable generating function, and it is found that p//A, p//B space decomposes into four regions of distinct qualitative properties. The asymptotic probability of the walk being to the right of the moving boundary is obtained explicitly in three of the four regions. However, analysis in the fourth region is a sensitive function of the denominator of the rational fraction gamma , and so we conclude with a number of special cases which can be solved in closed form.

AB - We study a random walk with nearest neighbor transitions on a one-dimensional lattice. The walk starts at the origin, as does a dividing line which moves with constant speed gamma , but the outward transition probabilities p//A and p//B differ on the right- and left-hand sides of the dividing line. This problem is solved formally by taking advantage of the analytical properties in the complex plane of an added variable generating function, and it is found that p//A, p//B space decomposes into four regions of distinct qualitative properties. The asymptotic probability of the walk being to the right of the moving boundary is obtained explicitly in three of the four regions. However, analysis in the fourth region is a sensitive function of the denominator of the rational fraction gamma , and so we conclude with a number of special cases which can be solved in closed form.

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M3 - Article

VL - 47

SP - 822

EP - 830

JO - SIAM Journal on Applied Mathematics

JF - SIAM Journal on Applied Mathematics

SN - 0036-1399

IS - 4

ER -