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Register a new userPatterson-Sullivan measures for point processes and the reconstruction of harmonic functions
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The Patterson-Sullivan construction is proved almost surely to recover every harmonic function in a certain Banach space from its values on the zero set of a Gaussian analytic function on the disk. The argument relies on the slow growth of variance for linear statistics of the concerned point process. As a corollary of reconstruction result in general abstract setting, Patterson-Sullivan reconstruction of harmonic functions is obtained in real and complex hyperbolic spaces of arbitrary dimension.
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The Patterson-Sullivan construction is proved almost surely to recover a Bergman function from its values on a random discrete subset sampled with the determinantal point process induced by the Bergman kernel on the unit ball $mathbb{D}_d$ in $mathbb{C}^d$. For super-critical weighted Bergman spaces, the interpolation is uniform when the functions range over the unit ball of the weighted Bergman space. As main results, we obtain a necessary and sufficient condition for interpolation of a fixed pluriharmonic function in the complex hyperbolic space of arbitrary dimension (cf. Theorem 1.4 and Theorem 4.11); optimal simultaneous uniform interpolation for weighted Bergman spaces (cf. Theorem 1.8, Proposition 1.9 and Theorem 4.13); strong simultaneous uniform interpolation for weighted harmonic Hardy spaces (cf. Theorem 1.11 and Theorem 4.15); and establish the impossibility of the uniform simultaneous interpolation for the Bergman space $A^2(mathbb{D}_d)$ on $mathbb{D}_d$ (cf. Theorem 1.12 and Theorem 6.7).
The gamma kernels are a family of projection kernels $K^{(z,z)}=K^{(z,z)}(x,y)$ on a doubly infinite $1$-dimensional lattice. They are expressed through Eulers gamma function and depend on two continuous parameters $z,z$. The gamma kernels initially arose from a model of random partitions via a limit transition. On the other hand, these kernels are closely related to unitarizable representations of the Lie algebra $mathfrak{su}(1,1)$. Every gamma kernel $K^{(z,z)}$ serves as a correlation kernel for a determinantal measure $M^{(z,z)}$, which lives on the space of infinite point configurations on the lattice. We examine chains of kernels of the form $$ ldots, K^{(z-1,z-1)}, ; K^{(z,z)},; K^{(z+1,z+1)}, ldots, $$ and establish the following hierarchical relations inside any such chain: Given $(z,z)$, the kernel $K^{(z,z)}$ is a one-dimensional perturbation of (a twisting of) the kernel $K^{(z+1,z+1)}$, and the one-point Palm distributions for the measure $M^{(z,z)}$ are absolutely continuous with respect to $M^{(z+1,z+1)}$. We also explicitly compute the corresponding Radon-Nikodym derivatives and show that they are given by certain normalized multiplicative functionals.
For a determinantal point process induced by the reproducing kernel of the weighted Bergman space $A^2(U, omega)$ over a domain $U subset mathbb{C}^d$, we establish the mutual absolute continuity of reduced Palm measures of any order provided that the domain $U$ contains a non-constant bounded holomorphic function. The result holds in all dimensions. The argument uses the $H^infty(U)$-module structure of $A^2(U, omega)$. A corollary is the quasi-invariance of our determinantal point process under the natural action of the group of compactly supported diffeomorphisms of $U$.
For a Pfaffian point process we show that its Palm measures, its normalised compositions with multiplicative functionals, and its conditional measures with respect to fixing the configuration in a bounded subset are Pfaffian point processes whose kernels we find explicitly.
The paper analyzes risk assessment for cash flows in continuous time using the notion of convex risk measures for processes. By combining a decomposition result for optional measures, and a dual representation of a convex risk measure for bounded cd processes, we show that this framework provides a systematic approach to the both issues of model ambiguity, and uncertainty about the time value of money. We also establish a link between risk measures for processes and BSDEs.
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