  We consider certain random matrix eigenvalue dynamics, akin to Dyson Brownian
motion, introduced by Rider and Valko. We show that from every initial
condition, including ones involving coinciding coordinates, the dynamics,
enhanced with more information, converge on path-space to an
infinite-dimensional Feller-continuous diffusion process. We show that the
limiting diffusion solves an infinite-dimensional system of stochastic
differential equations (ISDE) with logarithmic interaction. Moreover, we show
convergence in the long-time limit of the infinite-dimensional dynamics
starting from any initial condition to the equilibrium measure, given by the
inverse points of the Bessel determinantal point process. As far as we can
tell, this is: (a) the first path-space convergence result of random matrix
dynamics starting from every initial condition to an infinite-dimensional
Feller diffusion, (b) the first construction of solutions to an ISDE with
logarithmic interaction from every initial condition for which the singular
drift term can be defined at time $0$, (c) the first convergence to equilibrium
result from every initial condition for an ISDE of this kind. The argument
splits into two parts. The first part builds on the method of intertwiners
introduced and developed by Borodin and Olshanski. The main new ingredients are
a uniform, in a certain sense, approximation theorem of the spectrum of a
family of random matrices indexed by an infinite-dimensional space and an
extension of the method of intertwiners to deal with convergence to
equilibrium. The second part introduces a new approach towards convergence of
the singular drift term in the dynamics and for showing non-intersection of the
limiting paths via certain ``characteristic polynomials" associated to the
process. We believe variations of it will be applicable to other
infinite-dimensional dynamics coming from random matrices.
