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… 
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
...
...