Zero temperature limit for the Brownian directed polymer among Poissonian disasters

TL;DR

This paper investigates a continuum directed polymer model in a Poissonian disaster environment at zero temperature, proving the existence and continuity of the free energy where it was previously unestablished.

Contribution

It establishes the existence and continuity of the free energy for the zero-temperature limit of the Brownian directed polymer in Poissonian disasters, a problem unresolved in prior research.

Findings

Proved the existence of free energy at zero temperature.

Demonstrated the continuity of free energy at zero temperature.

Extended understanding of the model's behavior in the zero-temperature regime.

Abstract

We study a continuum model of directed polymer in random environment. The law of the polymer is defined as the Brownian motion conditioned to survive among space-time Poissonian disasters. This model is well-studied in the positive temperature regime. However, at zero-temperature, even the existence of the free energy has not been proved. In this article, we prove that the free energy exists and is continuous at zero-temperature.