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To a non-negative Borel measure dm(x) on an interval with suitable boundary conditions on the end points, we can associate a generalized differential operator A== -and a strong Markov process X on the support dm(x) dx of dm generated by the operator A. The measure dm is often called a string and the process X a generalized diffusion, also a quasi-diffusion or a gap diffusion, with the speed measure dm(x), cf. [9] for details. Let 0>^>/12>--be the eigenvalues of A= -and let ;?(r, x, y) be dm(x) dx the transition probability density of X with respect to dm(x). It was shown by M.G. Krein [10] and H.P. Mckean-D.B. Ray [12] that

Some asymptotic estimates of transition probability densities for generalized diffusion processes with self-similar speed measures