The continuum directed polymer in L\'evy Noise

TL;DR

This paper constructs a continuum directed polymer model in space-time Le9vy noise, focusing on b5-stable noises, to describe the scaling limit of discrete polymers in heavy-tail environments, and explores its properties and relation to stochastic heat equations.

Contribution

It introduces a novel continuum polymer model driven by Le9vy noise, extending the framework to arbitrary dimensions with specific conditions, and connects it to stochastic heat equations with multiplicative Le9vy noise.

Findings

Constructed a continuum polymer model in Le9vy noise environments.

Analyzed properties and dimension-dependent conditions of the model.

Established relation to stochastic heat equations with multiplicative Le9vy noise.

Abstract

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 L\'evy 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 L\'evy noise.