Results 1  10
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29
On some exponential functionals of Brownian motion
 Adv. Appl. Prob
, 1992
"... Abstract: This is the second part of our survey on exponential functionals of Brownian motion. We focus on the applications of the results about the distributions of the exponential functionals, which have been discussed in the first part. Pricing formula for call options for the Asian options, expl ..."
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Cited by 97 (10 self)
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Abstract: This is the second part of our survey on exponential functionals of Brownian motion. We focus on the applications of the results about the distributions of the exponential functionals, which have been discussed in the first part. Pricing formula for call options for the Asian options, explicit expressions for the heat kernels on hyperbolic spaces, diffusion processes in random environments and extensions of Lévy’s and Pitman’s theorems are discussed.
GENERATING FUNCTIONAL IN CFT AND EFFECTIVE ACTION FOR TWODIMENSIONAL QUANTUM GRAVITY ON HIGHER GENUS RIEMANN SURFACES
, 1996
"... We formulate and solve the analog of the universal Conformal Ward Identity for the stressenergy tensor on a compact Riemann surface of genus g> 1, and present a rigorous invariant formulation of the chiral sector in the induced twodimensional gravity on higher genus Riemann surfaces. Our const ..."
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Cited by 10 (5 self)
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We formulate and solve the analog of the universal Conformal Ward Identity for the stressenergy tensor on a compact Riemann surface of genus g> 1, and present a rigorous invariant formulation of the chiral sector in the induced twodimensional gravity on higher genus Riemann surfaces. Our construction of the action functional uses various double complexes naturally associated with a Riemann surface, with computations that are quite similar to descent calculations in BRST cohomology theory. We also provide an interpretation of the action functional in terms of the geometry of different fiber spaces over the Teichmüller space of compact Riemann surfaces of genus g> 1.
CYCLE INTEGRALS OF THE JFUNCTION AND MOCK MODULAR FORMS
"... In this paper we construct certain mock modular forms of weight 1/2 whose Fourier coefficients are cycle integrals of the modular jfunction and whose shadows are weakly holomorphic forms of weight 3/2. As an application we construct through a Shimuratype lift a holomorphic function that transforms ..."
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Cited by 7 (5 self)
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In this paper we construct certain mock modular forms of weight 1/2 whose Fourier coefficients are cycle integrals of the modular jfunction and whose shadows are weakly holomorphic forms of weight 3/2. As an application we construct through a Shimuratype lift a holomorphic function that transforms with a rational period function having poles at certain real quadratic integers. This function yields a real quadratic analogue of the Borcherds product.
Holomorphic factorization of determinants of Laplacians on Riemann surfaces and a higher genus generalization of Kronecker’s first limit formula
 GEOM. FUNCT. ANAL
, 2006
"... For a family of compact Riemann surfaces Xt of genus g> 1, parameterized by the Schottky space Sg, we define a natural basis of H 0 (Xt, ω n Xt) which varies holomorphically with t and generalizes the basis of normalized abelian differentials of the first kind for n = 1. We introduce a holomorph ..."
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Cited by 7 (0 self)
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For a family of compact Riemann surfaces Xt of genus g> 1, parameterized by the Schottky space Sg, we define a natural basis of H 0 (Xt, ω n Xt) which varies holomorphically with t and generalizes the basis of normalized abelian differentials of the first kind for n = 1. We introduce a holomorphic function F(n) on Sg which generalizes the classical product ∏∞ m=1 (1 − qm) 2 for n = 1 and g = 1. We prove the holomorphic factorization formula det ′ ∆n det Nn
Sum formula for Kloosterman sums and the fourth moment of the Dedekind zetafunction over the Gaussian number field. Functiones et Approximatio
"... Abstract. We prove the Kloosterman–Spectral sum formula for PSL2(Z[i])\PSL2(C), and apply it to derive an explicit spectral expansion for the fourth power moment of the Dedekind zeta function of the Gaussian number field. Our sum formula, Theorem 13.1, allows the extension of the spectral theory of ..."
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Cited by 6 (4 self)
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Abstract. We prove the Kloosterman–Spectral sum formula for PSL2(Z[i])\PSL2(C), and apply it to derive an explicit spectral expansion for the fourth power moment of the Dedekind zeta function of the Gaussian number field. Our sum formula, Theorem 13.1, allows the extension of the spectral theory of Kloosterman sums to all algebraic number fields. 1.
Cusps and the family hyperbolic metric
 Duke Math. J
, 2007
"... The hyperbolic metric for the punctured unit disc in the Euclidean plane is singular at the origin. A renormalization of the metric at the origin is provided by the Euclidean metric. For Riemann surfaces there is a unique germ for the isometry class of a complete hyperbolic metric at a cusp. The ren ..."
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Cited by 5 (2 self)
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The hyperbolic metric for the punctured unit disc in the Euclidean plane is singular at the origin. A renormalization of the metric at the origin is provided by the Euclidean metric. For Riemann surfaces there is a unique germ for the isometry class of a complete hyperbolic metric at a cusp. The renormalization for the punctured unit disc provides a renormalization for a hyperbolic metric at a cusp. For a holomorphic family of punctured Riemann surfaces the family of (co)tangent spaces along a puncture defines a tautological holomorphic line bundle over the base of the family. The Hermitian connection and Chern form for the renormalized metric are determined. Connections to the work of M. Mirzakhani, L. Takhtajan and P. Zograf, and intersection numbers for the moduli space of punctured Riemann surfaces studied by E. Witten are presented. 1 Comparing cusps The renormalization of a hyperbolic metric at a cusp is introduced. The setting is used to present an intrinsic norm for the germ of a holomorphic map at a cusp. A compact Riemann surface R having punctures and negative Euler characteristic has a complete hyperbolic metric, [Ahl73]. The geometry of a cusp of a hyperbolic metric is standard. From the uniformization theorem for a puncture p there is a distinguished local conformal coordinate with z(p) = 0 and the metric locally given by the germ of ds 2 =
INTEGRAL TRACES OF SINGULAR VALUES OF WEAK MAASS FORMS
"... ABSTRACT. We define traces associated to a weakly holomorphic modular form f of arbitrary negative even integral weight and show that these traces appear as coefficients of certain weakly holomorphic forms of halfintegral weight. If the coefficients of f are integral, then these traces are integral ..."
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Cited by 4 (3 self)
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ABSTRACT. We define traces associated to a weakly holomorphic modular form f of arbitrary negative even integral weight and show that these traces appear as coefficients of certain weakly holomorphic forms of halfintegral weight. If the coefficients of f are integral, then these traces are integral as well. We obtain a negative weight analogue of the classical Shintani lift and give an application to a generalization of the Shimura lift. 1.
On the arithmetic selfintersection numbers of the dualizing . . .
, 2009
"... We study the arithmetic selfintersection number of the dualizing sheaf on arithmetic surfaces with respect to morphisms of a particular kind. We obtain upper bounds for the arithmetic selfintersection number of the dualizing sheaf on minimal regular models of the modular curves associated with co ..."
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Cited by 2 (0 self)
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We study the arithmetic selfintersection number of the dualizing sheaf on arithmetic surfaces with respect to morphisms of a particular kind. We obtain upper bounds for the arithmetic selfintersection number of the dualizing sheaf on minimal regular models of the modular curves associated with congruence subgroups Γ0(N) with square free level, as well as for the modular curves X(N) and the Fermat curves with prime exponent.