The paper establishes local and global well-posedness for the 2D stochastic heat and damped wave equations with exponential nonlinearity in the ranges β²<1.37π (heat, any sign), β²<4π (heat, defocusing), and β²<0.86π (wave, defocusing), with invariance of the associated Gibbs measures.
Stochastic Ricci Flow on Compact Surfaces
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abstract
In this paper we introduce the stochastic Ricci flow (SRF) in two spatial dimensions. The flow is symmetric with respect to a measure induced by Liouville Conformal Field Theory. Using the theory of Dirichlet forms, we construct a weak solution to the associated equation of the area measure on a flat torus, in the full "$L^1$ regime" $\sigma< \sigma_{L^1}=2\sqrt\pi$ where $\sigma$ is the noise strength. We also describe the main necessary modifications needed for the SRF on general compact surfaces, and list some open questions.
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2019 1verdicts
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On the parabolic and hyperbolic Liouville equations
The paper establishes local and global well-posedness for the 2D stochastic heat and damped wave equations with exponential nonlinearity in the ranges β²<1.37π (heat, any sign), β²<4π (heat, defocusing), and β²<0.86π (wave, defocusing), with invariance of the associated Gibbs measures.