The continuum directed polymer in Lévy Noise

Quentin Berger (1, 2) , Hubert Lacoin

1 LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation
2 SU - Sorbonne Université

We present in this paper the construction of a continuum directed polymer model in an environment given by space-time L\'evy noise. One of the main objectives of this construction is to describe the scaling limit of discrete directed polymer in an heavy-tail environment and for this reason we put special emphasis on the case of $\alpha$-stable noises with $\alpha \in (1,2)$. Our construction can be performed in arbitrary dimension, provided that the Lévy measure satisfies specific (and dimension dependent) conditions. We also discuss a few basic properties of the continuum polymer and the relation between this model and the Stochastic Heat Equation with multiplicative Lévy noise.

Domaines: Probabilités [math.PR]; Physique mathématique [math-ph]

Quentin Berger, Hubert Lacoin. The continuum directed polymer in Lévy Noise. Journal de l'École polytechnique — Mathématiques, 2022, ⟨10.5802/jep.182⟩. ⟨hal-02977537⟩