The intermediate disorder regime for Brownian directed polymers in Poisson environment

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We consider the Brownian directed polymer in Poissonian environment in dimension 1+1, under the so-called intermediate disorder regime (Alberts et al., 2014), which is a crossover regime between the strong and weak disorder regions. We show that, under a diffusive scaling involving different parameters of the system, the renormalized point-to-point partition function of the polymer converges in law to the solution of the stochastic heat equation with Gaussian multiplicative noise. The Poissonian environment provides a natural setting and strong tools, such as the Wiener–Itô chaos expansion (Last and Penrose, 2017), which, applied to the partition function, is the basic ingredient of the proof.