On the one-sided exit problem for stable processes in random scenery

Fabienne Castell (Aix-Marseille Université)
Nadine Guillotin-Plantard (Université de Lyon)
Françoise Pène (Université de Brest)
Bruno Schapira (Aix-Marseille Université)


We consider the one-sided exit problem for stable Lévy process in random scenery, that is the asymptotic behaviour for $T$ large of the probability $$\mathbb{P}\Big[ \sup_{t\in[0,T]} \Delta_t \leq 1\Big] $$ where $$\Delta_t = \int_{\mathbb{R}} L_t(x) \, dW(x).$$ Here $W=(W(x))_{x\in\mathbb{R}}$ is a two-sided standard real Brownian motion and $(L_t(x))_{x\in\mathbb{R},t\geq 0}$ the local time of a stable Lévy process with index $\alpha\in (1,2]$, independent from the process $W$. Our result confirms some physicists prediction by Redner and Majumdar.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 1-7

Publication Date: May 14, 2013

DOI: 10.1214/ECP.v18-2444


  • Aurzada, F.; Simon, T. Persistence probabilities & exponents (2012) Math arXiv:1203.6554.
  • Aurzada, Frank. On the one-sided exit problem for fractional Brownian motion. Electron. Commun. Probab. 16 (2011), 392--404. MR2831079
  • Bingham, N. H. Maxima of sums of random variables and suprema of stable processes. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 26 (1973), 273--296. MR0415780
  • Bolthausen, Erwin. A central limit theorem for two-dimensional random walks in random sceneries. Ann. Probab. 17 (1989), no. 1, 108--115. MR0972774
  • Bouchaud, J. P.; Georges, A.; Koplik, J.; Provata, A.; Redner, S. Superdiffusion in random velocity fields. Phys. Rev. Lett. 64, 2503 -- 2506.
  • Borodin, A. N. A limit theorem for sums of independent random variables defined on a recurrent random walk. (Russian) Dokl. Akad. Nauk SSSR 246 (1979), no. 4, 786--787. MR0543530
  • Borodin, A. N. Limit theorems for sums of independent random variables defined on a transient random walk. (Russian) Investigations in the theory of probability distributions, IV. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 85 (1979), 17--29, 237, 244. MR0535455
  • Deligiannidis, G.; Utev, S. A. Computation of the asymptotics of the variance of the number of self-intersections of stable random walks using the Wiener-Darboux theory. (Russian) Sibirsk. Mat. Zh. 52 (2011), no. 4, 809--822; translation in Sib. Math. J. 52 (2011), no. 4, 639--650 MR2883216
  • Kesten, H.; Spitzer, F. A limit theorem related to a new class of self-similar processes. Z. Wahrsch. Verw. Gebiete 50 (1979), no. 1, 5--25. MR0550121
  • Khoshnevisan, Davar; Lewis, Thomas M. A law of the iterated logarithm for stable processes in random scenery. Stochastic Process. Appl. 74 (1998), no. 1, 89--121. MR1624017
  • Le Doussal, Pierre. Diffusion in layered random flows, polymers, electrons in random potentials, and spin depolarization in random fields. J. Statist. Phys. 69 (1992), no. 5-6, 917--954. MR1192029
  • Majumdar, S. Persistence of a particle in the Matheron - de Marsily velocity field. Phys. Rev. E 68, 050101(R) (2003).
  • Matheron, G.; de Marsily G. Is transport in porous media always diffusive? A counterexample. Water Resources Res. 16 (1980), 901 -- 907. �mbridge Studies in Advanced Mathematics 100, Cambridge University Press, Cambridge, (2006), x+620 pp.
  • Molchan, G. M. On the maximum of fractional Brownian motion. (Russian) Teor. Veroyatnost. i Primenen. 44 (1999), no. 1, 111--115; translation in Theory Probab. Appl. 44 (2000), no. 1, 97--102 MR1751192
  • Redner, S. Invited Symposium Contribution: Superdiffusion in Random Velocity Fields. Proceedings of the Bar-Ilan Conference in Condensed-Matter Physics, Physica A 168, 551 (1990).
  • Redner, S. Survival Probability in a Random Velocity Field. Phys. Rev., E56, 4967 (1997).

Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.