Almost sure asymptotics for the local time of a diffusion in Brownian environment

@article{Diel2011AlmostSA,
  title={Almost sure asymptotics for the local time of a diffusion in Brownian environment},
  author={Roland Diel},
  journal={Stochastic Processes and their Applications},
  year={2011},
  volume={121},
  pages={2303-2330}
}
  • R. Diel
  • Published 2 March 2010
  • Mathematics
  • Stochastic Processes and their Applications
View PDF on arXiv
8 Citations
Highly Influential Citations
2
Background Citations
6
Methods Citations
1
Results Citations
1

Figures from this paper

8 Citations

Citation Type

Citation Type

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 (−κ∕2)-drifted Brownian potential for κ ≠ 0. We are interested in the maximum of its local time, and study its almost sure asymptotic behaviour,…
Renewal structure and local time for diffusions in random environment
We study a one-dimensional diffusion $X$ in a drifted Brownian potential $W\_\kappa$, with $ 0\textless{}\kappa\textless{}1$, and focus on the behavior of the local times $(\mathcal{L}(t,x),x)$ of…
Almost sure behavior for the local time of a diffusion in a spectrally negative Lévy environment
We study the almost sure asymptotic behavior of the supremum of the local time for a transient diffusion in a spectrally negative Levy environment. In particular, we link this behavior with the left…
Invariant distributions and scaling limits for some diffusions in time-varying random environments
We consider a family of one-dimensional diffusions, in dynamical Wiener mediums, which are random perturbations of the Ornstein–Uhlenbeck diffusion process. We prove quenched and annealed…
Localization and number of visited valleys for a transient diffusion in random environment
We consider a transient diffusion in a $(-\kappa/2)$-drifted Brownian potential $W_{\kappa}$ with $0<\kappa<1$. We prove its localization at time $t$ in the neighborhood of some random points…
Rough paths and 1d SDE with a time dependent distributional drift: application to polymers
Motivated by the recent advances in the theory of stochastic partial differential equations involving nonlinear functions of distributions, like the Kardar–Parisi–Zhang (KPZ) equation, we reconsider…
Nonlinear Integrals, Diffusion in Random Environments and Stochastic Partial Differential Equations
In this dissertation, we investigate various problems in the analysis of stochastic (partial) differential equations. A part of the dissertation introduces several notions of nonlinear integrations.…
Stochastic Differential Equation for Brox Diffusion

References

SHOWING 1-10 OF 33 REFERENCES
Limit Law of the Local Time for Brox’s Diffusion
We consider Brox’s model: a one-dimensional diffusion in a Brownian potential W. We show that the normalized local time process (L(t,mlog t+x)/t, x∈R), where mlog t is the bottom of the deepest…
First passage times and sojourn times for Brownian motion in space and the exact Hausdorff measure of the sample path
also with probability 1. This result is proved in ?5. The difficulty in proving lower bounds like (1.2) is that one has to consider all possible coverings of the path by small convex sets in the…
Localization of favorite points for diffusion in a random environment
Annealed tail estimates for a Brownian motion in a drifted Brownian potential
We study Brownian motion in a drifted Brownian potential. Kawazu and Tanaka [J. Math. Soc. Japan 49 (1997) 189-211 exhibited two speed regimes for this process, depending on the drift. They…
Tightness of localization and return time in random environment
A local time curiosity in random environment
The Local time of Simple Random Walk in Random Environment
Two integral tests are established, which characterize respectively Lévy's upper and lower classes for the local time of Sinai's simple random walk in random environment. The weak convergence of the…
The limits of Sinai's simple random walk in random environment
We study the sample path asymptotics of a class of recurrent diffusion processes with random potentials, including examples of Sinai's simple random walk in random environment and Brox's diffusion…
Valleys and the Maximum Local Time for Random Walk in Random Environment
AbstractLet ξ (n, x) be the local time at x for a recurrent one-dimensional random walk in random environment after n steps, and consider the maximum ξ*(n) = maxx ξ(n, x). It is known that lim sup…
Renewal property of the extrema and tree property of the excursion of a one-dimensional brownian motion
Let ~ be the space of excursions, i.e. of real continuous functions 0 o defined on an interval [0,~(~o)] (0 O, those excursions o with heights g(~o ) z h will be called h-excursions. An excursion or…
...
...