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We consider a directed polymer model in dimension $1+1$, where the disorderis given by the occupation field of a Poisson system of independent randomwalks on $\mathbb Z$. In a suitable continuum and weak disorder limit, we showthat the family of quenched partition functions of the directed polymerconverges to the Stratonovich solution of a multiplicative stochastic heatequation (SHE) with a Gaussian noise, whose space-time covariance is given bythe heat kernel.  In contrast to the case with space-time white noise where the solution of theSHE admits a Wiener-It\^{o} chaos expansion, we establish an $L^1$-convergentchaos expansions of iterated integrals generated by Picard iterations. Usingthis expansion and its discrete counterpart for the polymer partitionfunctions, the convergence of the terms in the expansion is proved viafunctional analytic arguments and heat kernel estimates. The Poisson randomwalk system is amenable to careful moment analysis, which is an important inputto our arguments.

Scaling limit of a directed polymer among a Poisson field of independent walks