Integrable solutions of the stationary Kolmogorov equation
Consider the second-order elliptic operator \[ L = \sum_{1 \leq i,j \leq d} a^{ij} \partial_{x_i} \partial_{x_j} + \sum_{1 \leq i \leq d} b^i \partial_{x_i} \] and the corresponding stationary Kolmogorov equation (SKE) \[ L^{\star} \mu = 0, \] where \(L^{\star}\) is the (formal) adjoint operator of \(L\) and \(\mu\) is a bounded (i.e., finite) signed measure. In this short but interesting paper, the authors give a (sufficient) condition under which the nontrivial solutions of the SKE are of constant sign and unique up to a constant factor (hence, under their condition, there is a unique solution which is a probability measure). Since their condition (see Theorem 1) cannot be easily verified, they also give two broader conditions, which employ Lyapunov functions. Furthermore, the paper contains some interesting examples and recalls some relevant known results.
scientific article; zbMATH DE number 6106449