Georgia Institute of Technology College of Sciences

We study a broad class of space–time continuous directed polymers in a Gaussian environment that is white in time and spatially correlated (Itô sense). Under Dalang’s condition, we prove key properties of the partition function—positivity, stationarity, scaling, homogeneity, and a Chapman–Kolmogorov relation—and establish pathwise regularity of the polymer (Hölder continuity and quadratic variation). We give a sharp singularity criterion: the polymer measure is singular w.r.t. Wiener measure iff the spectral measure has infinite total mass. Finally, for d≥3, we prove diffusive large-time behavior in the high-temperature regime, providing a unified framework for polymers driven by singular Gaussian noise.

Joint work with Cheng Ouyang (UIC), Samy Tindel (Purdue), and Panqiu Xia (Cardiff).