### Abstract

We consider the solution of the equation r(t) = W(r(t)), r(0) = r _{0} > 0 where W(·) is a fractional Brownian motion (f.B.m.) with the Hurst exponent α ∈ (0,1). We show that for almost all realizations of W(·) the trajectory reaches in finite time the nearest equilibrium point (i.e. zero of the f.B.m.) either to the right or to the left of r_{0}, depending on whether W(r_{0}) is positive or not. After reaching the equilibrium the trajectory stays in it forever. The problem is motivated by studying the separation between two particles in a Gaussian velocity field which satisfies a local self-similarity hypothesis. In contrast to the case when the forcing term is a Brownian motion (then an analogous statement is a consequence of the Markov property of the process) we show our result using as the principal tools the properties of time reversibility of the law of the f.B.m., see Lemma 2.4 below, and the small ball estimate of Molchan, Commun. Math. Phys. 205 (1999) 97-111.

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

Pages (from-to) | 553-565 |

Number of pages | 13 |

Journal | Journal of Statistical Physics |

Volume | 127 |

Issue number | 3 |

DOIs | |

State | Published - May 2007 |

### Fingerprint

### Keywords

- Fractional Brownian motion
- Passive tracer
- Two point separation function

### ASJC Scopus subject areas

- Physics and Astronomy(all)
- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Journal of Statistical Physics*,

*127*(3), 553-565. https://doi.org/10.1007/s10955-006-9270-0

**Finite time approach to equilibrium in a fractional Brownian velocity field.** / Horvai, Peter; Komorowski, Tomasz; Wehr, Jan.

Research output: Contribution to journal › Article

*Journal of Statistical Physics*, vol. 127, no. 3, pp. 553-565. https://doi.org/10.1007/s10955-006-9270-0

}

TY - JOUR

T1 - Finite time approach to equilibrium in a fractional Brownian velocity field

AU - Horvai, Peter

AU - Komorowski, Tomasz

AU - Wehr, Jan

PY - 2007/5

Y1 - 2007/5

N2 - We consider the solution of the equation r(t) = W(r(t)), r(0) = r 0 > 0 where W(·) is a fractional Brownian motion (f.B.m.) with the Hurst exponent α ∈ (0,1). We show that for almost all realizations of W(·) the trajectory reaches in finite time the nearest equilibrium point (i.e. zero of the f.B.m.) either to the right or to the left of r0, depending on whether W(r0) is positive or not. After reaching the equilibrium the trajectory stays in it forever. The problem is motivated by studying the separation between two particles in a Gaussian velocity field which satisfies a local self-similarity hypothesis. In contrast to the case when the forcing term is a Brownian motion (then an analogous statement is a consequence of the Markov property of the process) we show our result using as the principal tools the properties of time reversibility of the law of the f.B.m., see Lemma 2.4 below, and the small ball estimate of Molchan, Commun. Math. Phys. 205 (1999) 97-111.

AB - We consider the solution of the equation r(t) = W(r(t)), r(0) = r 0 > 0 where W(·) is a fractional Brownian motion (f.B.m.) with the Hurst exponent α ∈ (0,1). We show that for almost all realizations of W(·) the trajectory reaches in finite time the nearest equilibrium point (i.e. zero of the f.B.m.) either to the right or to the left of r0, depending on whether W(r0) is positive or not. After reaching the equilibrium the trajectory stays in it forever. The problem is motivated by studying the separation between two particles in a Gaussian velocity field which satisfies a local self-similarity hypothesis. In contrast to the case when the forcing term is a Brownian motion (then an analogous statement is a consequence of the Markov property of the process) we show our result using as the principal tools the properties of time reversibility of the law of the f.B.m., see Lemma 2.4 below, and the small ball estimate of Molchan, Commun. Math. Phys. 205 (1999) 97-111.

KW - Fractional Brownian motion

KW - Passive tracer

KW - Two point separation function

UR - http://www.scopus.com/inward/record.url?scp=34247268210&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34247268210&partnerID=8YFLogxK

U2 - 10.1007/s10955-006-9270-0

DO - 10.1007/s10955-006-9270-0

M3 - Article

VL - 127

SP - 553

EP - 565

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 3

ER -