## Abstract

We consider the linear stochastic recursion $$x_{i+1} = a_{i}x_{i}+b_{i}$$xi+1=aixi+bi where the multipliers $$a_i$$ai are random and have Markovian dependence given by the exponential of a standard Brownian motion and $$b_{i}$$bi are i.i.d. positive random noise independent of $$a_{i}$$ai. Using large deviations theory we study the growth rates (Lyapunov exponents) of the positive integer moments $$\lambda _q = \lim _{n\rightarrow \infty } \frac{1}{n} \log \mathbb {E}[(x_n)^q]$$λq=limn→∞1nlogE[^{(xn)q}] with $$q\in \mathbb {Z}_+$$q∈Z+. We show that the Lyapunov exponents $$\lambda _q$$λq exist, under appropriate scaling of the model parameters, and have non-analytic behavior manifested as a phase transition. We study the properties of the phase transition and the critical exponents using both analytic and numerical methods.

Original language | English (US) |
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Pages (from-to) | 1354-1388 |

Number of pages | 35 |

Journal | Journal of Statistical Physics |

Volume | 160 |

Issue number | 5 |

DOIs | |

State | Published - Sep 1 2015 |

## Keywords

- Critical exponent
- Large deviations
- Linear stochastic recursion
- Lyapunov exponent
- Phase transitions