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. more »« less
Cavero, J.; Hofmann, S.; Martell, J.M.(
, Transactions of the American Mathematical Society)
Let $\Omega\subset\re^{n+1}$, $n\ge 2$, be a 1-sided chord-arc domain, that is, a domain which satisfies interior Corkscrew and Harnack Chain conditions (these are respectively scale-invariant/quantitative versions of the openness and path-connectedness), and whose boundary $\partial\Omega$ is $n$-dimensional Ahlfors regular. Consider $L_0$ and $L$ two real symmetric divergence form elliptic operators and let $\omega_{L_0}$, $\omega_L$ be the associated elliptic measures. We show that if $\omega_{L_0}\in A_\infty(\sigma)$, where $\sigma=H^n\lfloor_{\partial\Omega}$, and $L$ is a perturbation of $L_0$ (in the sense that the discrepancy between $L_0$ and $L$ satisfies certain Carleson measure condition), then $\omega_L\in A_\infty(\sigma)$. Moreover, if $L$ is a sufficiently small perturbation of $L_0$, then one can preserve the reverse Hölder classes, that is, if for some $1
Li, Weilin; Liao, Wenjing; Fannjiang, Albert(
, IEEE Transactions on Information Theory)
The problem of imaging point objects can be formulated as estimation of an unknown atomic measure from its M+1 consecutive noisy Fourier coefficients. The standard resolution of this inverse problem is 1/M and super-resolution refers to the capability of resolving atoms at a higher resolution. When any two atoms are less than 1/M apart, this recovery problem is highly challenging and many existing algorithms either cannot deal with this situation or require restrictive assumptions on the sign of the measure. ESPRIT is an efficient method which does not depend on the sign of the measure. This paper provides an explicit error bound on the support matching distance of ESPRIT in terms of the minimum singular value of Vandermonde matrices. When the support consists of multiple well-separated clumps and noise is sufficiently small, the support error by ESPRIT scales like SRF2λ-2×Noise, where the Super-Resolution Factor (SRF) governs the difficulty of the problem and λ is the cardinality of the largest clump. Our error bound matches the min-max rate of a special model with one clump of closely spaced atoms up to a factor of M in the small noise regime, and therefore establishes the near-optimality of ESPRIT. Our theory is validated by numerical experiments. Keywords: Super-resolution, subspace methods, ESPRIT, stability, uncertainty principle.
Triggiani, Roberto; Wan, Xiang(
, Evolution Equations and Control Theory)
We consider the linear third order (in time) PDE known as the SMGTJ-equation, defined on a bounded domain, under the action of either Dirichlet or Neumann boundary control \begin{document}$ g $\end{document}. Optimal interior and boundary regularity results were given in [1], after [41], when \begin{document}$ g \in L^2(0, T;L^2(\Gamma)) \equiv L^2(\Sigma) $\end{document}, which, moreover, in the canonical case \begin{document}$ \gamma = 0 $\end{document}, were expressed by the well-known explicit representation formulae of the wave equation in terms of cosine/sine operators [19], [17], [24,Vol Ⅱ]. The interior or boundary regularity theory is however the same, whether \begin{document}$ \gamma = 0 $\end{document} or \begin{document}$ 0 \neq \gamma \in L^{\infty}(\Omega) $\end{document}, since \begin{document}$ \gamma \neq 0 $\end{document} is responsible only for lower order terms. Here we exploit such cosine operator based-explicit representation formulae to provide optimal interior and boundary regularity results with \begin{document}$ g $\end{document} "smoother" than \begin{document}$ L^2(\Sigma) $\end{document}, qualitatively by one unit, two units, etc. in the Dirichlet boundary case. To this end, we invoke the corresponding results for wave equations, as in [17]. Similarly for the Neumann boundary case, by invoking the corresponding results for the wave equation as in [22], [23], [37] for control smoother than \begin{document}$ L^2(0, T;L^2(\Gamma)) $\end{document}, and [44] for control less regular in space than \begin{document}$ L^2(\Gamma) $\end{document}. In addition, we provide optimal interior and boundary regularity results when the SMGTJ equation is subject to interior point control, by invoking the corresponding wave equations results [42], [24,Section 9.8.2].
It has been recently established in David and Mayboroda (Approximation of green functions and domains with uniformly rectifiable boundaries of all dimensions.arXiv:2010.09793) that on uniformly rectifiable sets the Green function is almost affine in the weak sense, and moreover, in some scenarios such Green function estimates are equivalent to the uniform rectifiability of a set. The present paper tackles a strong analogue of these results, starting with the “flagship degenerate operators on sets with lower dimensional boundaries. We consider the elliptic operators$$L_{\beta ,\gamma } =- {\text {div}}D^{d+1+\gamma -n} \nabla $$associated to a domain$$\Omega \subset {\mathbb {R}}^n$$with a uniformly rectifiable boundary$$\Gamma $$of dimension$$d < n-1$$, the now usual distance to the boundary$$D = D_\beta $$given by$$D_\beta (X)^{-\beta } = \int _{\Gamma } |X-y|^{-d-\beta } d\sigma (y)$$for$$X \in \Omega $$, where$$\beta >0$$and$$\gamma \in (-1,1)$$. In this paper we show that the Green functionGfor$$L_{\beta ,\gamma }$$, with pole at infinity, is well approximated by multiples of$$D^{1-\gamma }$$, in the sense that the function$$\big | D\nabla \big (\ln \big ( \frac{G}{D^{1-\gamma }} \big )\big )\big |^2$$satisfies a Carleson measure estimate on$$\Omega $$. We underline that the strong and the weak results are different in nature and, of course, at the level of the proofs: the latter extensively used compactness arguments, while the present paper relies on some intricate integration by parts and the properties of the “magical distance function from David et al. (Duke Math J, to appear).
Postle, Luke; Thomas, Robin(
, Transactions of the American Mathematical Society. Series B)
We develop a theory of linear isoperimetric inequalities for graphs on surfaces and apply it to coloring problems, as follows. Let $ G$ be a graph embedded in a fixed surface $ \Sigma $ of genus $ g$ and let $ L=(L(v):v\in V(G))$ be a collection of lists such that either each list has size at least five, or each list has size at least four and $ G$ is triangle-free, or each list has size at least three and $ G$ has no cycle of length four or less. An $ L$-coloring of $ G$ is a mapping $ \phi $ with domain $ V(G)$ such that $ \phi (v)\in L(v)$ for every $ v\in V(G)$ and $ \phi (v)\ne \phi (u)$ for every pair of adjacent vertices $ u,v\in V(G)$. We prove
if every non-null-homotopic cycle in $ G$ has length $ \Omega (\log g)$, then $ G$ has an $ L$-coloring,
if $ G$ does not have an $ L$-coloring, but every proper subgraph does (``$ L$-critical graph''), then $ \vert V(G)\vert=O(g)$,
if every non-null-homotopic cycle in $ G$ has length $ \Omega (g)$, and a set $ X\subseteq V(G)$ of vertices that are pairwise at distance $ \Omega (1)$ is precolored from the corresponding lists, then the precoloring extends to an $ L$-coloring of $ G$,
if every non-null-homotopic cycle in $ G$ has length $ \Omega (g)$, and the graph $ G$ is allowed to have crossings, but every two crossings are at distance $ \Omega (1)$, then $ G$ has an $ L$-coloring,
if $ G$ has at least one $ L$-coloring, then it has at least $ 2^{\Omega (\vert V(G)\vert)}$ distinct $ L$-colorings.
We show that the above assertions are consequences of certain isoperimetric inequalities satisfied by $ L$-critical graphs, and we study the structure of families of embedded graphs that satisfy those inequalities. It follows that the above assertions hold for other coloring problems, as long as the corresponding critical graphs satisfy the same inequalities.
Dubédat, Julien, and Shen, Hao. Stochastic Ricci Flow on Compact Surfaces. Retrieved from https://par.nsf.gov/biblio/10326369. International Mathematics Research Notices . Web. doi:10.1093/imrn/rnab015.
Dubédat, Julien, & Shen, Hao. Stochastic Ricci Flow on Compact Surfaces. International Mathematics Research Notices, (). Retrieved from https://par.nsf.gov/biblio/10326369. https://doi.org/10.1093/imrn/rnab015
@article{osti_10326369,
place = {Country unknown/Code not available},
title = {Stochastic Ricci Flow on Compact Surfaces},
url = {https://par.nsf.gov/biblio/10326369},
DOI = {10.1093/imrn/rnab015},
abstractNote = {Abstract In this paper we introduce the stochastic Ricci flow (SRF) in two spatial dimensions. The flow is symmetric with respect to a measure induced by Liouville conformal field theory. Using the theory of Dirichlet forms, we construct a weak solution to the associated equation of the area measure on a flat torus, in the full “$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.},
journal = {International Mathematics Research Notices},
author = {Dubédat, Julien and Shen, Hao},
}
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