Continuous state branching processes in random environment: The Brownian case

We consider continuous state branching processes that are perturbed by a Brownian motion. These processes are constructed as the unique strong solution of a stochastic differential equation. The long-term extinction and explosion behaviours are studied. In the stable case, the extinction and explosion probabilities are given explicitly. We find three regimes for the asymptotic behaviour of the explosion probability and, as in the case of branching processes in random environment, we find five regimes for the asymptotic behaviour of the extinction probability. In the supercritical regime, we study the process conditioned on eventual extinction where three regimes for the asymptotic behaviour of the extinction probability appear. Finally, the process conditioned on non-extinction and the process with immigration are given.Comment: New version, Theorem 1 is improved and a new regime appears in the supercritical cas

The lower envelope of positive self-similar Markov processes

We establish integral tests and laws of the iterated logarithm for the lower envelope of positive self-similar Markov processes at 0 and $+\infty$. Our proofs are based on the Lamperti representation and time reversal arguments. These results extend laws of the iterated logarithm for Bessel processes due to Dvoretsky and Erd\"{o}s, Motoo and Rivero

Asymptotic behaviour of exponential functionals of L\'evy processes with applications to random processes in random environment

Let $\xi=(\xi_t, t\ge 0)$ be a real-valued L\'evy process and define its associated exponential functional as follows $$I_t(\xi):=\int_0^t \exp\{-\xi_s\}{\rm d} s, \qquad t\ge 0.$$ Motivated by important applications to stochastic processes in random environment, we study the asymptotic behaviour of $$\mathbb{E}\Big[F\big(I_t(\xi)\big)\Big] \qquad \textrm{as}\qquad t\to \infty,$$ where $F$ is a non-increasing function with polynomial decay at infinity and under some exponential moment conditions on $\xi$. In particular, we find five different regimes that depend on the shape of the Laplace exponent of $\xi$. Our proof relies on a discretisation of the exponential functional $I_t(\xi)$ and is closely related to the behaviour of functionals of semi-direct products of random variables. We apply our main result to three {questions} associated to stochastic processes in random environment. We first consider the asymptotic behaviour of extinction and explosion for {stable} continuous state branching processes in a L\'evy random environment. Secondly, we {focus on} the asymptotic behaviour of the mean of a population model with competition in a L\'evy random environment and finally, we study the tail behaviour of the maximum of a diffusion process in a L\'evy random environment.Comment: arXiv admin note: text overlap with arXiv:1512.07691, arXiv:math/0511265 by other authors. Results are improve

Regional growth and regional imbalances: Spain and USA

Regional and national incomes are determined by spatial and non spatial phenomena. Relative elasticities in Spain and U.S.A. for income explanatory variables, which were derivated from space models estimates, are 35 y 65%, respectively. External spatial economies, locational inertia of investment and urban expenditure multiplier -all of them typical spatial variables- are all concepts explaining the way space cooperates to income generation. Non spatial economic analysis therefore does not pay attention to phenomena which explain about a third of the produced national wealth. Regional income imbalances have not disappeared after decades or even centuries of economic development. Although they have indeed decreased dramatically, this reduction stopped around 1960 in the U.S.A. and 1980 in Spain. Since then regional per capita income imbalances have remained almost constant, ranging from 60 to 75 % between poor and rich regions. Interregional technological transfers, internal economies of scale, the urban expenditure multiplier, and decreasing external spatial economies explain the interregional “catching-up”. External spatial economies, locational inertia of investment and the end of regional labour migration are the reasons explaining the present steadiness in spatial imbalances.

A Wiener-Hopf Type Factorization for the Exponential Functional of Levy Processes

For a L\'evy process $\xi=(\xi_t)_{t\geq0}$ drifting to $-\infty$, we define the so-called exponential functional as follows $${\rm{I}}_{\xi}=\int_0^{\infty}e^{\xi_t} dt.$$ Under mild conditions on $\xi$, we show that the following factorization of exponential functionals $${\rm{I}}_{\xi}\stackrel{d}={\rm{I}}_{H^-} \times {\rm{I}}_{Y}$$ holds, where, $\times$ stands for the product of independent random variables, $H^-$ is the descending ladder height process of $\xi$ and $Y$ is a spectrally positive L\'evy process with a negative mean constructed from its ascending ladder height process. As a by-product, we generate an integral or power series representation for the law of ${\rm{I}}_{\xi}$ for a large class of L\'evy processes with two-sided jumps and also derive some new distributional properties. The proof of our main result relies on a fine Markovian study of a class of generalized Ornstein-Uhlenbeck processes which is of independent interest on its own. We use and refine an alternative approach of studying the stationary measure of a Markov process which avoids some technicalities and difficulties that appear in the classical method of employing the generator of the dual Markov process