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}
}
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… 
View PDF on arXiv
13 Citations
Highly Influential Citations
1
Background Citations
6
Methods Citations
2
Results Citations
1

13 Citations

Metric growth dynamics in Liouville quantum gravity
We consider the metric growth in Liouville quantum gravity (LQG) for γ ∈ (0, 2). We show that a process associated with the trace of the free field on the boundary of a filled LQG ball is stationary,…
The elliptic stochastic quantization of some two dimensional Euclidean QFTs
We study a class of elliptic SPDEs with additive Gaussian noise on $\mathbb{R}^2 \times M$, with $M$ a $d$-dimensional manifold equipped with a positive Radon measure, and a real-valued non linearity…
A probabilistic approach of Liouville field theory
In this article, we present the Liouville field theory, which was introduced in the eighties in physics by Polyakov as a model for fluctuating metrics in 2D quantum gravity, and outline recent…
Stochastic quantization associated with the $$\exp (\Phi )_2$$ exp ( Φ )
We consider a quantum field model with exponential interactions on the two-dimensional torus, which is called the $$\exp (\Phi )_{2}$$ exp ( Φ ) 2 -quantum field model or Høegh-Krohn’s model. In the…
A priori bounds for quasi-linear SPDEs in the full sub-critical regime
This paper is concerned with quasi-linear parabolic equations driven by an additive forcing ξ ∈ C, in the full subcritical regime α ∈ (0, 1). We are inspired by Hairer’s regularity structures,…
Stochastic quantization associated with the $$\exp (\Phi )_2$$-quantum field model driven by space-time white noise on the torus
We consider a quantum field model with exponential interactions on the two-dimensional torus, which is called the $\exp (\Phi)_{2}$-quantum field model or Hoegh-Krohn's model. In the present paper,…
On the Parabolic and Hyperbolic Liouville Equations
We study the two-dimensional stochastic nonlinear heat equation (SNLH) and stochastic damped nonlinear wave equation (SdNLW) with an exponential nonlinearity $$\lambda \beta e^{\beta u }$$ λ β…
Liouville quantum gravity from random matrix dynamics
We establish the first connection between 2 d Liouville quantum gravity and natural dynamics of random matrices. In particular, we show that if ( U t ) is a Brownian motion on the unitary group at…
Dynamical Fractional and Multifractal Fields
Motivated by the modeling of three-dimensional fluid turbulence, we define and study a class of stochastic partial differential equations (SPDEs) that are randomly stirred by a spatially smooth and…
Liouville Quantum Gravity with Matter Central Charge in (1, 25): A Probabilistic Approach
There is a substantial literature concerning Liouville quantum gravity (LQG) in two dimensions with conformal matter field of central charge $${{\mathbf {c}}}_{\mathrm M} \in (-\infty ,1]$$ c M ∈ ( -…
...
...

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
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
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
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
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
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
  • F. OttoH. Weber
  • 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…
...
...