Stochastic Ricci Flow on Compact Surfaces

@article{Dubedat2019StochasticRF,
title={Stochastic Ricci Flow on Compact Surfaces},
author={Julien Dub'edat and Hao Shen},
journal={arXiv: Probability},
year={2019}
}
• Published 24 April 2019
• Mathematics
• arXiv: Probability
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…
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References

SHOWING 1-10 OF 83 REFERENCES
The Ricci flow on the 2-sphere
The classical uniformization theorem, interpreted differential geomet-rically, states that any Riemannian metric on a 2-dimensional surface ispointwise conformal to a constant curvature metric. Thus
Stochastic Heat Equations with Values in a Manifold via Dirichlet Forms
• Mathematics
SIAM J. Math. Anal.
• 2020
In this paper, we prove the existence of martingale solutions to the stochastic heat equation taking values in a Riemannian manifold, which admits Wiener (Brownian bridge) measure on the Riemannian
Stochastic heat equations for infinite strings with values in a manifold
• Mathematics
• 2018
In the paper, we construct conservative Markov processes corresponding to the martingale solutions to the stochastic heat equation on R + \mathbb {R}^+ or R \mathbb {R}
Stochastic differential equations in infinite dimensions: solutions via Dirichlet forms
• Mathematics
• 1991
SummaryUsing the theory of Dirichlet forms on topological vector spaces we construct solutions to stochastic differential equations in infinite dimensions of the type $$dX_t = dW_t + \beta (X_t )dt$$
Ergodicity for the Stochastic Quantization Problems on the 2D-Torus
• Mathematics
• 2016
In this paper we study the stochastic quantization problem on the two dimensional torus and establish ergodicity for the solutions. Furthermore, we prove a characterization of the $${\Phi^4_2}$$Φ24
The Calabi metric for the space of Kähler metrics
Given any closed Kähler manifold we define, following an idea by Calabi (Bull. Am. Math. Soc. 60:167–168, 1954), a Riemannian metric on the space of Kähler metrics regarded as an infinite dimensional
Liouville quantum gravity on the annulus
• G. Remy
• Mathematics
Journal of Mathematical Physics
• 2018
In this work we construct Liouville quantum gravity on an annulus in the complex plane. This construction is aimed at providing a rigorous mathematical framework to the work of theoretical physicists
Renormalization Group and Stochastic PDEs
We develop a renormalization group (RG) approach to the study of existence and uniqueness of solutions to stochastic partial differential equations driven by space-time white noise. As an example, we
Geometric stochastic heat equations
• Mathematics
• 2019
We consider a natural class of $\mathbf{R}^d$-valued one-dimensional stochastic PDEs driven by space-time white noise that is formally invariant under the action of the diffeomorphism group on
Quasilinear SPDEs via Rough Paths
• Mathematics
Archive for Rational Mechanics and Analysis
• 2018
AbstractWe are interested in (uniformly) parabolic PDEs with a nonlinear dependence of the leading-order coefficients, driven by a rough right hand side. For simplicity, we consider a space-time