Francis Comets, Nobuo Yoshida
Brownian Directed Polymers in Random Environment
(365K, .pdf)

ABSTRACT.  We study the thermodynamics of a continuous model of 
directed polymers in random environment. 
The environment is given by a space-time Poisson 
point process, whereas the polymer is 
defined in terms of the Brownian motion. We mainly discuss: 
(i) The normalized partition function, its positivity in the limit 
which characterizes the phase diagram of the model. 
%and the equivalence 
%of its decay rate with some natural localization properties of the path; 
(ii) The existence of quenched Lyapunov exponent, its positivity, and 
its agreement with the annealed Lyapunov exponent; 
(iii) The longitudinal fluctuation of the free energy, some of its relations 
with the overlap between replicas and 
with the transversal fluctuation of the path. \\ 
The model considered here, enables us to use stochastic calculus, with 
respect to both Brownian motion and Poisson process, leading to 
 handy formulas for fluctuations analysis 
and qualitative properties of the phase diagram. We also relate our model 
to some formulation of the Kardar-Parisi-Zhang equation, more precisely, 
the stochastic heat equation. 
Our fluctuation results are interpreted 
as bounds on various exponents and provide a circumstantial 
evidence of super-diffusivity in dimension one. 
We also obtain an almost sure large deviation principle for the 
polymer measure.