Multiplicative chaos theory studies random measures defined on metric spaces whose density can be locally written as a product of independent random weights. It originates from the study of energy dissipation in fully developed turbulence by Kolmogorov in 1940s. The study was developed by Mandelbrot in 1970s, suggesting that the random measures under consideration should be obtained through a limiting procedure and should be carried by random fractal sets. The mathematically rigorous theory of multiplicative chaos was built by Kahane and Peyriere in 1976 and then by Kahane in 1985. Since then much progress has been made to multiplicative chaos theory, and numbers of connections and applications have been found to other area of mathematics and natural science, such as branching random walks, percolations, random polymers, quantum gravity and mathematical finance.
The problems I am working on include:
- The finer study of Gaussian multiplicative chaos, which is closely related to the Liouville quantum gravity and 2d statistical physics.
- Ergodic theory and dynamical system regarding measure-valued sequences obtained from multiplicative chaos, motivated by the study of projected and sliced percolation on fractal sets.
- Geometric properties of random fractals related to multiplicative chaos, such as the Hausdorff dimension and multifractal analysis of the random conformal welding curves using multiplicative chaos measures.