Singularities in d-dimensional Langevin equations with anisotropic multiplicative noise and lack of self-adjointness in the corresponding Schrodinger equation

Abstract: In this paper we analyse d-dimensional Langevin equations in Ito representation characterised by anisotropic multiplicative noise, composed by the superposition of an isotropic tensorial component and a radial one, and a radial power law drift term. This class of model is relevant in many contexts ranging from vortex stochastic dynamics, passive scalar transport in fully developed turbulence and second order phase transitions from active to absorbing states. The focus of the paper is the system behaviour around the singularity at vanishing distance depending on the model parameters. This can vary from regular boundary to naturally repulsive or attractive ones. The work develops in the following steps: (i) introducing a mapping that disentangle the radial dynamics from the angular one, with the first characterised by additive noise and either logarithmic or power law potential, and the second one being simply a free isotropic Brownian motion on the unitary sphere surface; (ii) applying the Feller's - Van Kampen's classification of singularities for continuous Markov processes; (iii) developing an Exponent Hunter Method to find the small distance scaling behaviour of the solutions; (iv) building a mapping into a well studied Schrodinger equation with singular potential in order to build a bridge between Feller's theory of singular boundaries of a continuous Markov process and the problem of self-adjointness of the Hamiltonian operator in quantum theory.