57,317 research outputs found
Almost sure asymptotics for the maximum local time in Brownian environment
We study the asymptotic behaviour of the maximum local time L*(t) of the
Brox's process, the diffusion in Brownian environment. Shi proved that the
maximum speed of L*(t) is surprisingly, at least t log(log(log t)) whereas in
the discrete case it is t. We show here that t log(log(log t)) is the proper
rate and we prove that for the minimum speed the rate is the same as in the
discrete case namely t/log(log(log t))
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 the unique solvability of
one-dimensional stochastic differential equations, the drift of which is a
distribution, by means of rough paths theory. Existence and uniqueness are
established in the weak sense when the drift reads as the derivative of a
H{\"o}lder continuous function. Regularity of the drift part is investigated
carefully and a related stochastic calculus is also proposed, which makes the
structure of the solutions more explicit than within the earlier framework of
Dirichlet processes
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;m_(log t) + x)=t; x \in
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
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