A new topological approach to the L∞-uniqueness of operators and the L1-uniqueness of Fokker–Planck equations

The usual semigroups of kernels on a Polish space E are in general not strongly continuous on L∞(E,μ) with respect to the norm topology. We introduce a new topology on L∞(E,μ) such that they become C0-semigroups for which we can establish a simplified Hille–Yosida theorem. The new topology will allow us to introduce the uniqueness of pre-generator on L∞(E,μ) which turns out to be equivalent to the L1-uniqueness of the associated Fokker–Planck equation among many others, and it is intimately related with the Liouville properties for L1-harmonic functions. The uniqueness of several second order elliptic differential operators in L∞ are studied: (1) one-dimensional diffusion operators a(x)f″+b(x)f′; (2) Schrödinger operators −(1/2)Δ+V; (3) multi-dimensional diffusion generator (1/2)Δ+β⋅∇.