Results 1  10
of
23
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 ..."
Abstract

Cited by 98 (9 self)
 Add to MetaCart
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
 Comm. Math. Phys
, 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 constr ..."
Abstract

Cited by 12 (7 self)
 Add to MetaCart
. 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 Teichmuller space of compact Riemann surfaces of genus g ? 1. 1. Introduction Conformal symmetry in two dimensions, according to Belavin, Polyakov, and Zamolodchikov [8], is generated by the holomorphic and antiholomorphic components T(z) and ¯ T(¯z) of the stressenergy tensor of a Conformal Field Theory. These components satisfy the Operator Product Expansions [8, 15] T(z)...
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 holomorphic ..."
Abstract

Cited by 7 (0 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 6 (4 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 5 (2 self)
 Add to MetaCart
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 =
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 ..."
Abstract

Cited by 5 (4 self)
 Add to MetaCart
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.
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 ..."
Abstract

Cited by 3 (2 self)
 Add to MetaCart
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.
Extension of the weilpetersson connection
, 2007
"... Convexity properties of WeilPetersson geodesics on the Teichmüller space of punctured Riemann surfaces are investigated. A normal form is presented for the WeilPetersson LeviCivita connection for pinched hyperbolic metrics. The normal form is used to establish approximation of geodesics in bounda ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
Convexity properties of WeilPetersson geodesics on the Teichmüller space of punctured Riemann surfaces are investigated. A normal form is presented for the WeilPetersson LeviCivita connection for pinched hyperbolic metrics. The normal form is used to establish approximation of geodesics in boundary spaces. Considerations are combined to establish convexity along WeilPetersson geodesics of the functions the distance between horocycles for a hyperbolic metric. 1
Product formulas on a unitary group in three variables
"... In the present paper, we consider the eigenvalue problems which concern a differential equation (1.1) Lkf(z1, z2) = λf(z1, z2). Here f is a function on the Siegel domain of type II ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
In the present paper, we consider the eigenvalue problems which concern a differential equation (1.1) Lkf(z1, z2) = λf(z1, z2). Here f is a function on the Siegel domain of type II
HARMONIC DIFFERENTIALS AND INFINITE GEODESIC JOINING TWO PUNCTURES ON A RIEMANN SURFACE
"... Abstract. Let M be a hyperbolic Riemann surface of finite volume. The harmonic dual form to an infinite geodesic joining two punctures on M is obtained in two different ways. First of all, using the degeneration of hyperbolic Eisenstein series, it is made explicit in terms of these. Secondly, genera ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
Abstract. Let M be a hyperbolic Riemann surface of finite volume. The harmonic dual form to an infinite geodesic joining two punctures on M is obtained in two different ways. First of all, using the degeneration of hyperbolic Eisenstein series, it is made explicit in terms of these. Secondly, generalizing the construction of Kudla and Millson to the case of an infinite geodesic joining two punctures, we give an automorphic realization of this harmonic form.