The maximum of the local time of a diffusion process in a drifted Brownian potential

We consider a one-dimensional diffusion process $X$ in a $(-\kappa/2)$-drifted Brownian potential for $\kappa\neq 0$. We are interested in the maximum of its local time, and study its almost sure asymptotic behaviour, which is proved to be different from the behaviour of the maximum local time of the transient random walk in random environment. We also obtain the convergence in law of the maximum local time of $X$ under the annealed law after suitable renormalization when $\kappa \geq 1$. Moreover, we characterize all the upper and lower classes for the hitting times of $X$, in the sense of Paul L\'evy, and provide laws of the iterated logarithm for the diffusion $X$ itself. To this aim, we use annealed technics.Comment: 38 pages, new version, merged with hal-00013040 (arXiv:math/0511053), with some additional result

Spawning period of Mediterranean marine fishes

We collected all available information (i.e. international and local journals, conference proceedings, theses, technical reports) on the spawning season (n = 511 stocks, 168 species), gonadosomatic index (n = 237 stocks, 81 species) and sex ratio (n = 97 stocks, 68 species) of Mediterranean marine fish. The 511 stocks represented 20 orders (most were Perciformes, 283 stocks) and 65 families (most were Sparidae: 17 species and 63 stocks). Overall, 346 stocks (128 species) spawned between April and August, 139 stocks (60 species) between September and March, while the remaining 26 stocks (13 species) were all-year-round spawners. In addition, 174 stocks (34.1%) were characterised by an extended (> 4 months) spawning season, but, for most stocks (332 stocks, 64.4%), spawning duration ranged from 2 to 4 months inclusive. Regardless of the onset and the duration of spawning, the spawning period of 284 and 287 stocks included June and July, respectively, indicating that most Mediterranean species are summer spawners. Female gonadosomatic index ranged between 0.06 and 37 (mean +/- A SE = 8.55 +/- A 0.647, n = 95) and was significantly higher (t-test: t = 5.58, P < 0.001) than the corresponding male one, which ranged between 0.06 and 30 (mean +/- A SE = 4.21 +/- A 0.431, n = 95). Congeneric species that occupied the same area and share the same requirements exhibited successive and non-overlapping spawning (e.g. Sparidae in the Adriatic Sea, Mugilidae in the Ionian Sea and Tunisian waters). The knowledge of the spawning period coupled with information on spawning and nursery grounds and detailed knowledge of mating systems, social interactions, maturity and fecundity may be very useful for fisheries management