2008/09/12 by Pierre Andreoletti, Andreoletti, Pierre, Roland Diel +1 · 1 citation
Biochemistry, Genetics and Molecular Biology · Economics, Econometrics and Finance · Mathematics · #Diffusion and Search Dynamics #FOS: Mathematics #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics
paper · doi:10.48550/arxiv.0809.2195
openalex publication_date 2008/09/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider Brox's model: a one-dimensional diffusion in a Brownian potential W. We show that the normalized local time process (L(t;m_(log t) + x)=t; x ∈ R), where m_(log t) is the bottom of the deepest valley reached by the process before time t, behaves asymptotically like a process which only depends on W. As a consequence, we get the weak convergence of the local time to a functional of two independent three-dimensional Bessel processes and thus the limit law of the supremum of the normalized local time. These results are discussed and compared to the discrete time and space case which same questions have been solved recently by N. Gantert, Y. Peres and Z. Shi.