The Logarithmic Laplacian on General Graphs

Abstract: We are the first to present a Bochner integral representation of the logarithmic Laplacian on weighted graphs and to derive, under the assumption of stochastic completeness, its explicit pointwise form. For weighted lattice graphs with uniformly positive vertex measures, we establish sharp upper and lower bounds for the associated log kernel, prove that the logarithmic Laplacian fails to be bounded on l2, and offer an alternative derivation of the above formula. Moreover, we compute the Fourier multipliers of both the fractional and logarithmic Laplacians, and derive the exact large time behavior and off diagonal asymptotics of their heat kernels, including all sharp asymptotic constants.