We provide several reasons as to why we think our definition is natural; in particular, we show that, for a convolution diffusion kernel on [R.sup.n], we achieve [alpha] = 1 in the discussion just above; i.e., we can recover (local) Euclidean distance to the "full" power 1.
The authors show that their decay assumption holds for semigroups arising in many different settings (for which suitable decay and continuity assumptions are made on diffusion kernels relative to an intrinsic metric D of the underlying space), and even for some examples of nonsymmetric diffusion kernels.
Wang, "The determination of the
diffusion kernel on fractals and fractional diffusion equation for transport phenomena in random media," Physics Letters A, vol.