Selfsimilar diffusions

This paper addresses symmetric Ito diffusions, with general position-dependent velocities and diffusion coefficients, that run over the real line. The paper establishes an explicit characterization of diffusions that are selfsimilar processes with arbitrary Hurst exponents. Specifically, for a given diffusion, the paper asserts that the four following statements are equivalent: (i) the diffusion's trajectories are selfsimilar; (ii) the diffusion's positions display a certain scaling property; (iii) the diffusion's first-passage times (FTPs) to the origin display a certain scaling property; (iv) the diffusion's velocities and diffusion coefficients admit certain power-law forms. For diffusions with constant diffusion coefficients, the paper further asserts that: selfsimilarity holds if and only if the underpinning potential is logarithmic, in which case the Hurst exponent is. The positions of selfsimilar diffusions are shown to have non-Gaussian statistics with the following features: densities that are either unimodal and explosive, or unimodal, or bimodal; and tails that are either light', or Gamma, or 'heavy'. The FTPs to the origin of selfsimilar diffusions are shown to have inverse-Gamma statistics with infinite means and infinite moments. The results presented in this paper unveil a class of Markovian diffusions, whose trajectories are fractal objects, and whose non-Gaussian statistics can be custom designed via three parameters: a velocity parameter, a diffusion parameter, and a Hurst exponent.