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We consider directed polymers in a random environment. Under some mild assumptions on the environment, we prove equivalence between the decay rate of the partition function and some natural localization properties of the path; some quantitative estimates of the decay of the partition function in one or two dimensions, or at sufficiently low temperature; and the existence of quenched free energy.In particular, we generalize to general environments the results recently obtained by Carmona and Hu for a Gaussian environment. Our approach is based on martingale decomposition and martingale analysis. It leads to a natural, asymptotic relation between the partition function, and the probability that two polymers in the same environment, but otherwise independent, end up at the same point.

In this paper, we consider directed polymers in random environment with discrete space and time. For transverse dimension at least equal to 3, we prove that diffusivity holds for the path in the full weak disorder region, i.e., where the partition function differs from its annealed value only by a non-vanishing factor. Deep inside this region, we also show that the quenched averaged energy has fluctuations of order 1. In complete generality (arbitrary dimension and temperature), we prove monotonicity of the phase diagram in the temperature. Short Title. Diffusive Directed Polymers

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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. (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.Short Title: Brownian Polymers