Nonlinear Fokker-Planck equations with fractional Laplacian and McKean-Vlasov SDEs with Lévy noise (Q6582357)

This work is concerned with the existence of mild solutions to the nonlinear Fokker-Planck equation (NFPE) given by \( u_t + (-\Delta)^s \beta(u) + \text{div}(Db(u)u) = 0 \) in \( (0, \infty) \times \mathbb{R}^d \), with the initial condition \( u(0, x) = u_0(x) \), where \( x \in \mathbb{R}^d \). Here, \( \beta : \mathbb{R} \to \mathbb{R} \), \( D : \mathbb{R}^d \to \mathbb{R}^d \), \( d \geq 2 \), and \( b : \mathbb{R} \to \mathbb{R} \) are functions specified later, while \( (-\Delta)^s \), with \( 0 < s < 1 \), is the fractional Laplace operator. The uniqueness of Schwartz distributional solutions is also proven under suitable assumptions on the diffusion and drift terms. As applications, the weak existence and uniqueness of solutions to McKean-Vlasov equations with Lévy noise, as well as the Markov property for their laws, are established. The dual of the Schwartz test function space \( S := S(\mathbb{R}^d) \), denoted \( S' := S'(\mathbb{R}^d) \), is utilized in the analysis.