Dice Question Streamline Icon: https://streamlinehq.com

Heat kernel of Liouville Brownian motion

Determine the precise structure of the heat kernel of Liouville Brownian motion associated with Gaussian multiplicative chaos M^g_γ on a compact Riemann surface (Σ,g) for γ in (0,2), including its quantitative behavior and asymptotics.

Information Square Streamline Icon: https://streamlinehq.com

Background

Gaussian multiplicative chaos measures define multifractal random geometries. The associated diffusion, Liouville Brownian motion, has been constructed, but detailed heat-kernel properties remain elusive.

Understanding the heat kernel would yield fine analytic control of diffusion on Liouville quantum gravity surfaces and connect probabilistic and geometric aspects of the theory.

References

The measure $Mg_\gamma$ is the prototype of multifractal random geometry: its associated Brownian motion has been introduced in , but the precise structure of its heat kernel is still unresolved despite progress .

— Two Decades of Probabilistic Approach to Liouville Conformal Field Theory (2509.21053 - Rhodes et al., 25 Sep 2025) in Section 3 (Probabilistic foundations)