McKean-Vlasov flights without moments: well-posedness, propagation of chaos and a critical threshold for law-dependent jumps

Abstract

We study McKean-Vlasov equations driven by an isotropic $2s$-stable L{é}vy noise, $s\in(0,1)$, whose solutions may have no finite mean, with coefficients that are Lipschitz with respect to a bounded Wasserstein-type distance. When the jump noise is additive, we prove strong well-posedness for every initial law and every $s$, without any moment assumption, and propagation of chaos at the rate $N^{-1/2}$ for kernel interactions. When the size of the jumps depends on the law, we prove the same results for $s>1/2$, and we show by an example that uniqueness fails for every $s<1/2$. The value $s=1/2$ thus separates well-posedness from non-uniqueness; the critical case itself remains open. These results answer, for stable noise, the question of uniqueness without moments raised by Jourdain, M{é}l{é}ard and Woyczy{ń}ski and by Cavallazzi.