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124
Equidistribution of small points, rational dynamics, and potential theory
 Ann. Inst. Fourier (Grenoble
, 2006
"... Abstract. Given a dynamical system associated to a rational function ϕ(T) on P 1 of degree at least 2 with coefficients in a number field k, we show that for each place v of k, there is a unique probability measure µϕ,v on the Berkovich space P 1 Berk,v /Cv such that if {zn} is a sequence of points ..."
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Cited by 46 (7 self)
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Abstract. Given a dynamical system associated to a rational function ϕ(T) on P 1 of degree at least 2 with coefficients in a number field k, we show that for each place v of k, there is a unique probability measure µϕ,v on the Berkovich space P 1 Berk,v /Cv such that if {zn} is a sequence of points in P 1 (k) whose ϕcanonical heights tend to zero, then the zn’s and their Galois conjugates are equidistributed with respect to µϕ,v. In the archimedean case, µϕ,v coincides with the wellknown canonical measure associated to ϕ. This theorem generalizes a result of BakerHsia [BH] when ϕ(z) is a polynomial. The proof uses a polynomial lift F (x, y) = (F1(x, y), F2(x, y)) of ϕ to construct a twovariable ArakelovGreen’s function gϕ,v(x, y) for each v. The measure µϕ,v is obtained by taking the Berkovich space Laplacian of gϕ,v(x, y), using a theory developed in [RB]. The other ingredients in the proof are (i) a potentialtheoretic energy minimization principle which says that � � gϕ,v(x, y) dν(x)dν(y) is uniquely minimized over all probability measures ν on P 1 Berk,v when ν = µϕ,v, and (ii) a formula for homogeneous transfinite diameter of the vadic filled Julia set KF,v ⊂ C 2 v in terms of the resultant Res(F) of F1 and F2. The resultant formula, which generalizes a formula of DeMarco [DeM], is proved using results
FRACTIONAL CAUCHY PROBLEMS ON BOUNDED DOMAINS
, 2009
"... Fractional Cauchy problems replace the usual firstorder time derivative by a fractional derivative. This paper develops classical solutions and stochastic analogues for fractional Cauchy problems in a bounded domain D⊂R d with Dirichlet boundary conditions. Stochastic solutions are constructed via ..."
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Cited by 41 (10 self)
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Fractional Cauchy problems replace the usual firstorder time derivative by a fractional derivative. This paper develops classical solutions and stochastic analogues for fractional Cauchy problems in a bounded domain D⊂R d with Dirichlet boundary conditions. Stochastic solutions are constructed via an inverse stable subordinator whose scaling index corresponds to the order of the fractional time derivative. Dirichlet problems corresponding to iterated Brownian motion in a bounded domain are then solved by establishing a correspondence with the case of a halfderivative in time.
Efficient mechanism design
, 1998
"... We study Bayesian mechanism design in situations where agents ’ information may be multidimensional, concentrating on mechanisms that lead to efficient allocations. Our main result is that a generalization of the wellknown VickreyClarkeGroves mechanism maximizes the planner’s “revenue ” among al ..."
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Cited by 40 (0 self)
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We study Bayesian mechanism design in situations where agents ’ information may be multidimensional, concentrating on mechanisms that lead to efficient allocations. Our main result is that a generalization of the wellknown VickreyClarkeGroves mechanism maximizes the planner’s “revenue ” among all efficient mechanisms. This result is then used to study multiple object auctions in situations where bidders have privately known “demand curves” and extended to include situations with complementarities across objects or externalities across bidders. We also illustrate how the main result may be used to analyze the possibility of allocating both private and public goods efÞciently when budget balance considerations are important. The generalized VCG mechanism, therefore, serves to unify many results in mechansim design theory. 1
Brownian subordinators and fractional Cauchy problems
, 2007
"... Abstract. A Brownian time process is a Markov process subordinated to the absolute value of an independent onedimensional Brownian motion. Its transition densities solve an initial value problem involving the square of the generator of the original Markov process. An apparently unrelated class of p ..."
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Cited by 35 (14 self)
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Abstract. A Brownian time process is a Markov process subordinated to the absolute value of an independent onedimensional Brownian motion. Its transition densities solve an initial value problem involving the square of the generator of the original Markov process. An apparently unrelated class of processes, emerging as the scaling limits of continuous time random walks, involve subordination to the inverse or hitting time process of a classical stable subordinator. The resulting densities solve fractional Cauchy problems, an extension that involves fractional derivatives in time. In this paper, we will show a close and unexpected connection between these two classes of processes, and consequently, an equivalence between these two families of partial differential equations. 1.
Scaling For A Random Polymer
, 1994
"... Let Q fi n be the law of the nstep random walk on Z d obtained by weighting simple random walk with a factor e \Gammafi for every selfintersection (DombJoyce model of `soft polymers'). It was proved by Greven and den Hollander (1993) that in d = 1 and for every fi 2 (0; 1) there exist ` ..."
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Cited by 32 (7 self)
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Let Q fi n be the law of the nstep random walk on Z d obtained by weighting simple random walk with a factor e \Gammafi for every selfintersection (DombJoyce model of `soft polymers'). It was proved by Greven and den Hollander (1993) that in d = 1 and for every fi 2 (0; 1) there exist ` (fi) 2 (0; 1) and ¯ fi 2 f¯ 2 l 1 (N) : k¯k l 1 = 1; ¯ ? 0g such that under the law Q fi n as n !1: (i) ` (fi) is the limit empirical speed of the random walk; (ii) ¯ fi is the limit empirical distribution of the local times. A representation was given for ` (fi) and ¯ fi in terms of a largest eigenvalue problem for a certain family of N \Theta N matrices. In the present paper we use this representation to prove the following scaling result as fi # 0: (i) fi \Gamma 1 3 ` (fi) ! b ; (ii) fi \Gamma 1 3 ¯ fi (d\Deltafi \Gamma 1 3 e) ! L 1 j (\Delta). The limits b 2 (0; 1) and j 2 fj 2 L 1 (R + ) : kjk L 1 = 1; j ? 0g are identified in terms of...
Formal GNS Construction and States in Deformation Quantization
, 1996
"... In this paper we develop a method of constructing Hilbert spaces and the representation of the formal algebra of quantum observables in deformation quantization which is an analog of the wellknown GNS construction for complex C ∗algebras: in this approach the corresponding positive linear function ..."
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Cited by 31 (3 self)
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In this paper we develop a method of constructing Hilbert spaces and the representation of the formal algebra of quantum observables in deformation quantization which is an analog of the wellknown GNS construction for complex C ∗algebras: in this approach the corresponding positive linear functionals (‘states’) take their values not in the field of complex numbers, but in (a suitable extension field of) the field of formal complex Laurent series in the formal parameter. By using the algebraic and topological properties of these fields we prove that this construction makes sense and show in physical examples that standard representations as the Bargmann and Schrödinger representation come out correctly, both formally and in a suitable convergence scheme. For certain Hamiltonian functions (contained in the Gel’fand ideal of the positive functional) a formal solution to the timedependent Schrödinger equation is shown to exist. Moreover, we show that for every Kähler manifold equipped with the Fedosov star product of Wick type all the classical delta functionals are positive and give rise to some formal Bargmann representation of the deformed algebra.
Tail behaviour of the busy period of a GI/GI/1 queue with subexponential service times
, 2004
"... This paper considers ..."
Sequentially Optimal Mechanisms
 Review of Economic Studies
, 2006
"... We characterize the revenue maximizing mechanism in a twoperiod model. A risk neutral seller owns one unit of a durable good and faces a risk neutral buyer whose valuation is private information. The seller has all the bargaining power; she designs an institution to sell the object at t=0 but she c ..."
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Cited by 21 (2 self)
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We characterize the revenue maximizing mechanism in a twoperiod model. A risk neutral seller owns one unit of a durable good and faces a risk neutral buyer whose valuation is private information. The seller has all the bargaining power; she designs an institution to sell the object at t=0 but she cannot commit not to change the institution at t=1 if trade does not occur at t=0. The seller’s objective is to pick the revenue maximizing mechanism. We show that if the probability density function of the buyer’s valuation satisfies certain conditions, the optimal mechanism is to post a price in each period. Previous work has examined price dynamics when the seller behaves sequentially rationally. We provide a reason for the seller’s choice to post a price even though she can use infinitely many other possible institutions: posted price selling is the optimal strategy in the sense that it maximizes the seller’s revenues. Keywords: mechanism design, optimal auctions, sequential rationality. JEL Classification Codes: C72, D44, D82. 1
On the rigidity of discrete isometry groups of negatively curved spaces
 COMMENTARII MATHEMATICI HELVETICI
, 1997
"... We prove an ergodic rigidity theorem for discrete isometry groups of CAT(−1) spaces. We give explicit examples of divergence isometry groups with infinite covolume in the case of trees, piecewise hyperbolic 2polyhedra, hyperbolic BruhatTits buildings and rank one symmetric spaces. We prove that t ..."
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Cited by 21 (3 self)
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We prove an ergodic rigidity theorem for discrete isometry groups of CAT(−1) spaces. We give explicit examples of divergence isometry groups with infinite covolume in the case of trees, piecewise hyperbolic 2polyhedra, hyperbolic BruhatTits buildings and rank one symmetric spaces. We prove that two negatively curved Riemannian metrics, with conical singularities of angles at least 2π, on a closed surface, with boundary map absolutely continuous with respect to the PattersonSullivan measures, are isometric. For that, we generalize J.P. Otal’s result to prove that a negatively curved Riemannian metric, with conical singularities of angles at least 2π, on a closed surface, is determined, up to isometry, by its marked length spectrum.