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This paper introduces a class of non-linear and continuous-time opiniondynamics model with additive noise and state dependent interaction ratesbetween agents. The model features interaction rates which are proportional toa negative power of opinion distances. We establish a non-local partialdifferential equation for the distribution of opinion distances and use Mellintransforms to provide an explicit formula for the stationary solution of thelatter, when it exists. Our approach leads to new qualitative and quantitativeresults on this type of dynamics. To the best of our knowledge these Mellintransform results are the first quantitative results on the equilibria ofopinion dynamics with distance-dependent interaction rates. The closed formexpressions for this class of dynamics are obtained for the two agent case.However the results can be used in mean-field models featuring several agentswhose interaction rates depend on the empirical average of their opinions. Thetechnique also applies to linear dynamics, namely with a constant interactionrate, on an interaction graph.

On the Steady State of Continuous Time Stochastic Opinion Dynamics with Power Law Confidence