64 research outputs found

    Brownian couplings, convexity, and shy-ness

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    Benjamini, Burdzy, and Chen (2007) introduced the notion of a shy coupling: a coupling of a Markov process such that, for suitable starting points, there is a positive chance of the two component processes of the coupling staying at least a given positive distance away from each other for all time. Among other results, they showed that no shy couplings could exist for reflected Brownian motions in C-2 bounded convex planar domains whose boundaries contain no line segments. Here we use potential-theoretic methods to extend this Benjamini et al. (2007) result (a) to all bounded convex domains (whether planar and smooth or not) whose boundaries contain no line segments, (b) to all bounded convex planar domains regardless of further conditions on the boundary

    Heat kernel estimates for general boundary problems

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    We show that not feeling the boundary estimates for heat kernels hold for any non-negative self-adjoint extension of the Laplace operator acting on vector-valued compactly supported functions on a domain in R d Rd . They are therefore valid for any choice of boundary condition and we show that the implied constants can be chosen independent of the self-adjoint extension. The method of proof is very general and is based on fi nite propagation speed estimates and explicit Fourier Tauberian theorems obtained by Y. Safarov

    On certain general integral operators of analytic functions

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    In this paper, we obtain new sufficient conditions for the operators Fα1,α2,...,αn,β(z)F_{\alpha_1,\alpha_2,...,\alpha_n,\beta}(z) and Gα1,α2,...,αn,β(z)G_{\alpha_1,\alpha_2,...,\alpha_n,\beta}(z) to be univalent in the open unit disc U\mathcal{U}, where the functions f1,f2,...,fnf_1, f_2,..., f_n belong to the classes S(a,b)S^*(a, b) and K(a,b)\mathcal{K}(a, b). The order of convexity for the operators Fα1,α2,...,αn,β(z)F_{\alpha_1,\alpha_2,...,\alpha_n,\beta}(z) and Gα1,α2,...,αn,β(z)G_{\alpha_1,\alpha_2,...,\alpha_n,\beta}(z) is also determined. Furthermore, and for β=1\beta= 1, we obtain sufficient conditions for the operators Fn(z)F_n(z) and Gn(z)G_n(z) to be in the class K(a,b)\mathcal{K}(a, b). Several corollaries and consequences of the main results are also considered

    Starlike Functions of Complex Order with Respect to Symmetric Points Defined Using Higher Order Derivatives

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    In this paper, we introduce and study a new subclass of multivalent functions with respect to symmetric points involving higher order derivatives. In order to unify and extend various well-known results, we have defined the class subordinate to a conic region impacted by Janowski functions. We focused on conic regions when it pertained to applications of our main results. Inclusion results, subordination property and coefficient inequality of the defined class are the main results of this paper. The applications of our results which are extensions of those given in earlier works are presented here as corollaries

    Local spectral asymptotics and heat kernel bounds for Dirac and Laplace operators

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    In this dissertation we study non-negative self-adjoint Laplace type operators acting on smooth sections of a vector bundle. First, we assume base manifolds are compact, boundaryless, and Riemannian. We start from the Fourier integral operator representation of half-wave operators, continue with spectral zeta functions, heat and resolvent trace asymptotic expansions, and end with the quantitative Wodzicki residue method. In particular, all of the asymptotic coefficients of the microlocalized spectral counting function can be explicitly given and clearly interpreted. With the auxiliary pseudo-differential operators ranging all smooth endomorphisms of the given bundle, we obtain certain asymptotic estimates about the integral kernel of heat operators. As applications, we study spectral asymptotics of Dirac type operators such as characterizing those for which the second coefficient vanishes. Next, we assume vector bundles are trivial and base manifolds are Euclidean domains, and study non-negative self-adjoint extensions of the Laplace operator which acts component-wise on compactly supported smooth functions. Using finite propagation speed estimates for wave equations and explicit Fourier Tauberian theorems obtained by Yuri Safarov, we establish the principle of not feeling the boundary estimates for the heat kernel of these operators. In particular, the implied constants are independent of self-adjoint extensions. As a by-product, we affirmatively answer a question about upper estimate for the Neumann heat kernel. Finally, we study some specific values of the spectral zeta function of two-dimensional Dirichlet Laplacians such as spectral determinant and Casimir energy. For numerical purposes we substantially improve the short-time Dirichlet heat trace asymptotics for polygons. This could be used to measure the spectral determinant and Casimir energy of polygons whenever the first several hundred or one thousand Dirichlet eigenvalues are known with high precision by other means
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