### Abstract

The notion of random motion in a potential well is elemental in the physical sciences and beyond. Quantitatively, this notion is described by reverting diffusions - asymptotically stationary diffusion processes which are simultaneously (i) driven toward a reversion level by a deterministic force, and (ii) perturbed off the reversion level by a random white noise. The archetypal example of reverting diffusions is the Ornstein-Uhlenbeck process, which is mean-reverting. In this paper we analyze reverting diffusions and establish that: (i) if the magnitude of the perturbing noise is constant then the diffusion's stationary density is unimodal and the diffusion is mode-reverting; (ii) if the magnitude of the perturbing noise is non-constant then, in general, neither is the diffusion's stationary density unimodal, nor is the diffusion mode-reverting. In the latter case we further establish a result asserting when unimodality and mode-reversion do hold. In particular, we demonstrate that the notion of mean-reversion, which is fundamental in economics and finance, is a misconception - as mean-reversion is an exception rather than the norm.

Original language | English (US) |
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Article number | 332001 |

Journal | Journal of Physics A: Mathematical and Theoretical |

Volume | 45 |

Issue number | 33 |

DOIs | |

State | Published - Aug 24 2012 |

### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Statistics and Probability
- Modeling and Simulation
- Mathematical Physics
- Physics and Astronomy(all)

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## Cite this

*Journal of Physics A: Mathematical and Theoretical*,

*45*(33), [332001]. https://doi.org/10.1088/1751-8113/45/33/332001