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An integral kernel approach to noise. (English) Zbl 0652.60068

Quantum probability and applications III, Proc. Conf., Oberwolfach/FRG 1987, Lect. Notes Math. 1303, 192-208 (1988).
[For the entire collection see Zbl 0627.00022.]
A stochastic calculus based on integral kernels is developed for the Wiener process using a convenient kernel notation for Fock spaces. sics, Proc. Taniguchi Int. Symp., Katata and Kyoto/Jap. 1985, 83-90 (1987).
[For the entire collection see Zbl 0633.00021.]
Let \(0>\lambda_ 1>\lambda_ 2\geq \lambda_ 3..\). be the eigenvalues of the problem \(\Delta f=\lambda f\) in [0,1], \(f(0)=f(1)=0\). A classical theorem of H. Weyl [Rend. Circ. Mat. Palermo 39, 1-50 (1915)] states (for this particular case): \(-\lambda_ n\sim cn^ 2\). In the paper \(\Delta\) is replaced by the generalized differential operator \(L=(d/dm)(d/dx)\), and the boundary condition by \[ f(0) \cos \alpha +(d/dx)f(0) \sin \alpha =0,\quad f(1) \cos \beta -(d/dx)f(1) \sin \beta =0 \] for some \(0\leq \alpha\), \(\beta\leq \pi\). The rate of growth of \(\lambda_ n\) is found for “self-similar measures” m.
A measure m on [0,1] is called self-similar [following J. E. Hutchinson, Indiana Univ. Math. J. 30, 713-747 (1981; Zbl 0598.28011)] for a given family \((S_ 1,...,S_ N)\) of contracting similarities \(S_ ix=r_ ix+b_ i\), \(0<r_ i<1\), mapping [0,1] into itself, and a family \((\rho_ 1,...,\rho_ N)\) of weights \(\rho_ i>0\), \(\sum^{N}_{i=1}\rho_ i=1\), if \(m=\sum^{N}_{i=1}\rho_ im\circ S_ i^{-1}\); in symbols \(m=m(S,\rho).\)
Theorem 3.6. \(-\lambda_ n\asymp n^{(1+s)/s}\) for large n, where s is from \[ \sum^{N}_{i=1}(\rho_ ir_ i)^{s/(1+s)}=1,\quad and\quad m=m(S,\rho). \] Under Hutchinson’s open set condition and for \(\rho_ i=r^ D_ i\) a corollary states \(-\lambda_ n\asymp n^{(1+D)/D},\) where D is the self-similarity dimension \((=\) Hausdorff dimension). Two examples consider the Cantor measure and the deRham measure.
Reviewer: U.Zähle

MSC:

60H99 Stochastic analysis
81P20 Stochastic mechanics (including stochastic electrodynamics)
60G99 Stochastic processes
60J60 Diffusion processes
35P15 Estimates of eigenvalues in context of PDEs
35P20 Asymptotic distributions of eigenvalues in context of PDEs