The Experts below are selected from a list of 50523 Experts worldwide ranked by ideXlab platform
Vincent Vargas - One of the best experts on this subject based on the ideXlab platform.
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Liouville brownian motion
Annals of Probability, 2016Co-Authors: Christophe Garban, Remi Rhodes, Vincent VargasAbstract:We construct a stochastic process, called the Liouville Brownian motion which we conjecture to be the scaling limit of random walks on large planar maps which are embedded in the euclidean plane or in the sphere in a conformal manner. Our construction works for all universality classes of planar maps satisfying $\gamma <\gamma_c=2$. In particular, this includes the interesting case of $\gamma=\sqrt{8/3}$ which corresponds to the conjectured scaling limit of large uniform planar $p$-angulations (with fixed $p\geq 3$). We start by constructing our process from some fixed point $x\in \R^2$ (or $x\in \S^2$). This amounts to changing the speed of a standard two-dimensional brownian motion $B_t$ depending on the local behaviour of the Liouville measure ''$M_\gamma(dz) = e^{\gamma X} dz$'' (where $X$ is a Gaussien Free Field, say on $\S^2$). A significant part of the paper focuses on extending this construction simultaneously to all points $x\in \R^2$ or $\S^2$ in such a way that one obtains a semi-group $P_\t$ (the Liouville semi-group). We prove that the associated Markov process is a Feller diffusion for all $\gamma<\gamma_c=2$ and that for $\gamma<\sqrt{2}$, the Liouville measure $M_\gamma$ is invariant under $P_\t$ (which in some sense shows that it is the right quantum gravity diffusion to consider). This Liouville Brownian motion enables us to give sense to part of the celebrated Feynman path integrals which are at the root of Liouville quantum gravity, the Liouville Brownian ones. Finally we believe that this work sheds some new light on the difficult problem of constructing a quantum metric out of the exponential of a Gaussian Free Field (see conjecture \ref{c.metric}).
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on the heat kernel and the dirichlet form of Liouville brownian motion
Electronic Journal of Probability, 2014Co-Authors: Remi Rhodes, Christophe Garban, Vincent VargasAbstract:In a previous work, a Feller process called Liouville Brownian motion on $\mathbb{R}^2$ has been introduced. It can be seen as a Brownian motion evolving in a random geometry given formally by the exponential of a (massive) Gaussian Free Field $e^{\gamma\, X}$ and is the right diffusion process to consider regarding $2d$-Liouville quantum gravity. In this note, we discuss the construction of the associated Dirichlet form, following essentially Fukushima, Oshima, and Takeda, and the techniques introduced in our previous work. Then we carry out the analysis of the Liouville resolvent. In particular, we prove that it is strong Feller, thus obtaining the existence of the Liouville heat kernel via a non-trivial theorem of Fukushima and al. One of the motivations which led to introduce the Liouville Brownian motion in our previous work was to investigate the puzzling Liouville metric through the eyes of this new stochastic process. In particular, the theory developed for example in Stollmann and Sturm, whose aim is to capture the "geometry" of the underlying space out of the Dirichlet form of a process living on that space, suggests a notion of distance associated to a Dirichlet form. More precisely, under some mild hypothesis on the regularity of the Dirichlet form, they provide a distance in the wide sense, called intrinsic metric, which is interpreted as an extension of Riemannian geometry applicable to non differential structures. We prove that the needed mild hypotheses are satisfied but that the associated intrinsic metric unfortunately vanishes, thus showing that renormalization theory remains out of reach of the metric aspect of Dirichlet forms.
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on the heat kernel and the dirichlet form of Liouville brownian motion
arXiv: Probability, 2013Co-Authors: Christophe Garban, Remi Rhodes, Vincent VargasAbstract:In \cite{GRV}, a Feller process called Liouville Brownian motion on $\R^2$ has been introduced. It can be seen as a Brownian motion evolving in a random geometry given formally by the exponential of a (massive) Gaussian Free Field $e^{\gamma X}$ and is the right diffusion process to consider regarding 2d-Liouville quantum gravity. In this note, we discuss the construction of the associated Dirichlet form, following essentially \cite{fuku} and the techniques introduced in \cite{GRV}. Then we carry out the analysis of the Liouville resolvent. In particular, we prove that it is strong Feller, thus obtaining the existence of the Liouville heat kernel via a non-trivial theorem of Fukushima and al. One of the motivations which led to introduce the Liouville Brownian motion in \cite{GRV} was to investigate the puzzling Liouville metric through the eyes of this new stochastic process. One possible approach was to use the theory developed for example in \cite{stollmann,sturm1,sturm2}, whose aim is to capture the "geometry" of the underlying space out of the Dirichlet form of a process living on that space. More precisely, under some mild hypothesis on the regularity of the Dirichlet form, they provide an intrinsic metric which is interpreted as an extension of Riemannian geometry applicable to non differential structures. We prove that the needed mild hypotheses are satisfied but that the associated intrinsic metric unfortunately vanishes, thus showing that renormalization theory remains out of reach of the metric aspect of Dirichlet forms.
A. M. Akhtyamov - One of the best experts on this subject based on the ideXlab platform.
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Degenerate Boundary Conditions for the Sturm-Liouville Problem on a Geometric Graph
Differential Equations, 2019Co-Authors: Victor Antonovich Sadovnichii, Ya. T. Sultanaev, A. M. AkhtyamovAbstract:We study the boundary conditions of the Sturm-Liouville problem posed on a star-shaped geometric graph consisting of three edges with a common vertex. We show that the Sturm-Liouville problem has no degenerate boundary conditions in the case of pairwise distinct edge lengths. However, if the edge lengths coincide and all potentials are the same, then the characteristic determinant of the Sturm-Liouville problem cannot be a nonzero constant and the set of Sturm-Liouville problems whose characteristic determinant is identically zero and whose spectrum accordingly coincides with the entire plane is infinite (a continuum). It is shown that, for one special case of the boundary conditions, this set consists of eighteen classes, each having from two to four arbitrary constants, rather than of two problems as in the case of the Sturm-Liouville problem on an interval.
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Inverse Sturm-Liouville Problem with Nonseparated Boundary Conditions on a Geometric Graph
Differential Equations, 2019Co-Authors: Victor Antonovich Sadovnichii, Ya. T. Sultanaev, A. M. AkhtyamovAbstract:The inverse Sturm-Liouville problem with nonseparated boundary conditions on a star-shaped geometric graph consisting of three edges with a common vertex is studied. It is shown that the Sturm-Liouville problem with general boundary conditions cannot be reconstructed uniquely from four spectra. A class of nonseparated boundary conditions is obtained for which two uniqueness theorems for the solution of the inverse Sturm-Liouville problem are proved. In the first theorem, the data used to reconstruct the Sturm-Liouville problem are the spectrum of the boundary value problem itself and the spectra of three auxiliary problems with separated boundary conditions. In the second theorem, instead of the spectrum of the problem itself, one only deals with five of its eigenvalues. It is shown that the Sturm-Liouville problem with these nonseparated boundary conditions can be reconstructed uniquely if three spectra of auxiliary problems and five eigenvalues of the problem itself are used as the reconstruction data. Examples of unique reconstruction of potentials and boundary conditions of the Sturm-Liouville problem posed on the graph under study are given.
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on the uniqueness of the solution of the inverse sturm Liouville problem with nonseparated boundary conditions on a geometric graph
Doklady Mathematics, 2018Co-Authors: V A Sadovnichy, Ya. T. Sultanaev, A. M. AkhtyamovAbstract:For the first time, the inverse Sturm–Liouville problem with nonseparated boundary conditions is studied on a star-shaped geometric graph with three edges. It is shown that the Sturm–Liouville problem with general boundary conditions cannot be uniquely reconstructed from four spectra. Nonseparated boundary conditions are found for which a uniqueness theorem for the solution of the inverse Sturm–Liouville problem is proved. The spectrum of the boundary value problem itself and the spectra of three auxiliary problems are used as reconstruction data. It is also shown that the Sturm–Liouville problem with these nonseparated boundary conditions can be uniquely recovered if three spectra of auxiliary problems are used as reconstruction data and only five of its eigenvalues are used instead of the entire spectrum of the problem.
Dumitru Baleanu - One of the best experts on this subject based on the ideXlab platform.
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representation of solutions for sturm Liouville eigenvalue problems with generalized fractional derivative
Chaos, 2020Co-Authors: Ramazan Ozarslan, Erdal Bas, Dumitru BaleanuAbstract:We analyze fractional Sturm-Liouville problems with a new generalized fractional derivative in five different forms. We investigate the representation of solutions by means of ρ-Laplace transform for generalized fractional Sturm-Liouville initial value problems. Finally, we examine eigenfunctions and eigenvalues for generalized fractional Sturm-Liouville boundary value problems. All results obtained are compared with simulations in detail under different α fractional orders and real ρ values.
Boris Dubrovin - One of the best experts on this subject based on the ideXlab platform.
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minimal Liouville gravity correlation numbers from douglas string equation
Journal of High Energy Physics, 2014Co-Authors: A A Belavin, Boris Dubrovin, Baur MukhametzhanovAbstract:We continue the study of (q, p) Minimal Liouville Gravity with the help of Douglas string equation. We generalize the results of [1,2], where Lee-Yang series (2, 2s + 1) was studied, to (3, 3s + p0) Minimal Liouville Gravity, where p0 = 1, 2. We demonstrate that there exist such coordinates τm,n on the space of the perturbed Minimal Liouville Gravity theories, in which the partition function of the theory is determined by the Douglas string equation. The coordinates τm,n are related in a non-linear fashion to the natural coupling constants λm,n of the perturbations of Minimal Lioville Gravity by the physical operators Om,n. We find this relation from the requirement that the correlation numbers in Minimal Liouville Gravity must satisfy the conformal and fusion selection rules. After fixing this relation we compute three- and four-point correlation numbers when they are not zero. The results are in agreement with the direct calculations in Minimal Liouville Gravity available in the literature [3, 4, 5].
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minimal Liouville gravity correlation numbers from douglas string equation
arXiv: High Energy Physics - Theory, 2013Co-Authors: A A Belavin, Boris Dubrovin, Baur MukhametzhanovAbstract:We continue the study of $(q,p)$ Minimal Liouville Gravity with the help of Douglas string equation. We generalize the results of \cite{Moore:1991ir}, \cite{Belavin:2008kv}, where Lee-Yang series $(2,2s+1)$ was studied, to $(3,3s+p_0)$ Minimal Liouville Gravity, where $p_0=1,2$. We demonstrate that there exist such coordinates $\tau_{m,n}$ on the space of the perturbed Minimal Liouville Gravity theories, in which the partition function of the theory is determined by the Douglas string equation. The coordinates $\tau_{m,n}$ are related in a non-linear fashion to the natural coupling constants $\lambda_{m,n}$ of the perturbations of Minimal Lioville Gravity by the physical operators $O_{m,n}$. We find this relation from the requirement that the correlation numbers in Minimal Liouville Gravity must satisfy the conformal and fusion selection rules. After fixing this relation we compute three- and four-point correlation numbers when they are not zero. The results are in agreement with the direct calculations in Minimal Liouville Gravity available in the literature \cite{Goulian:1990qr}, \cite{Zamolodchikov:2005sj}, \cite{Belavin:2006ex}.
Thomas E. St. George - One of the best experts on this subject based on the ideXlab platform.
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Linear Sturm-Liouville problems with Riemann- Stieltjes integral boundary conditions
Opuscula Mathematica, 2018Co-Authors: Qingkai Kong, Thomas E. St. GeorgeAbstract:We study second-order linear Sturm-Liouville problems involving general homogeneous linear Riemann-Stieltjes integral boundary conditions. Conditions are obtained for the existence of a sequence of positive eigenvalues with consecutive zero counts of the eigenfunctions. Additionally, we find interlacing relationships between the eigenvalues of such Sturm-Liouville problems and those of Sturm-Liouville problems with certain two-point separated boundary conditions.
