Weak uniqueness of Fokker-Planck equations with degenerate and bounded coefficients (Q964441)

Consider a stochastic differential equation \[ dX_t = b_t(X_t) dt + \sigma_t(X_t) dW_t \] with generator \[ L_t f (x) = \sum_i b^i_t(x) \partial_i f(x) + \frac{1}{2} \sum_{i,j}(\sigma \sigma^*)_{ij} \partial^2_{ij} f(x) \] and adjoint \(L^*_t\). The associated Fokker-Planck equation is \(\partial_t \mu_t = L^*_t \mu_t\) where \(\mu\) is a measure valued function. If \(\mu_t\) is absolutely continuous, its density satisfies the PDE \(\partial_t u_t = L^*_t u_t\) in the weak sense. Using this link between SDEs and Fokker-Planck equations, the authors prove that under some weak regularity assumptions a) the function-valued Fokker-Planck PDE has at most one solution in a suitable \(L^p\)-space b) the measure-valued Fokker-Planck equation has at most one solution.