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We consider a {\it{Gaussian multiplicative chaos}} (GMC) measure on theclassical Wiener space driven by a smoothened (Gaussian) space-time whitenoise. For $d\geq 3$ it was shown in \cite{MSZ16} that for small noiseintensity, the total mass of the GMC converges to a strictly positive randomvariable, while larger disorder strength (i.e., low temperature) forces thetotal mass to lose uniform integrability, eventually producing a vanishinglimit. Inspired by strong localization phenomena for log-correlated Gaussianfields and Gaussian multiplicative chaos in the finite dimensional Euclideanspaces (\cite{MRV16,BL18}), and related results for discrete directed polymers(\cite{V07,BC16}), we study the endpoint distribution of a Brownian path underthe {\it{renormalized}} GMC measure in this setting. We show that in the lowtemperature regime, the energy landscape of the system freezes and enters theso called {\it{glassy phase}} as the entire mass of the Ces\`aro average of theendpoint GMC distribution stays localized in few spatial islands, forcing theendpoint GMC to be {\it{asymptotically purely atomic}} (\cite{V07}). The methodof our proof is based on the translation-invariant compactification introducedin \cite{MV14} and a fixed point approach related to the cavity method fromspin glasses recently used in \cite{BC16} in the context of the directedpolymer model in the lattice.

Localization of the Gaussian multiplicative chaos in the Wiener space and the stochastic heat equation in strong disorder