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Phase transitions and effects of external noise on many body systems are oneof the main topics in physics. In mean field coupled nonlinear dynamicalstochastic systems driven by Brownian noise, various types of phase transitionsincluding nonequilibrium ones may appear. A Brownian motion is a special caseof L\'evy motion and the stochastic process based on the latter is analternative choice for studying cooperative phenomena in various fields.Recently, fractional Fokker-Planck equations associated with L\'evy noise haveattracted much attention and behaviors of systems with double-well potentialsubjected to L\'evy noise have been studied intensively. However, most of suchstudies have resorted to numerical computation. We construct an {\itanalytically solvable model} to study the occurrence of phase transitionsdriven by L\'evy stable noise.

Phase transitions driven by L\'evy stable noise: exact solutions and stability analysis of nonlinear fractional Fokker-Planck equations