  This paper introduces a class of non-linear and continuous-time opinion
dynamics model with additive noise and state dependent interaction rates
between agents. The model features interaction rates which are proportional to
a negative power of opinion distances. We establish a non-local partial
differential equation for the distribution of opinion distances and use Mellin
transforms to provide an explicit formula for the stationary solution of the
latter, when it exists. Our approach leads to new qualitative and quantitative
results on this type of dynamics. To the best of our knowledge these Mellin
transform results are the first quantitative results on the equilibria of
opinion dynamics with distance-dependent interaction rates. The closed form
expressions for this class of dynamics are obtained for the two agent case.
However the results can be used in mean-field models featuring several agents
whose interaction rates depend on the empirical average of their opinions. The
technique also applies to linear dynamics, namely with a constant interaction
rate, on an interaction graph.
