About semilinear low dimension Bessel PDEs

We prove existence and uniqueness of solutions of a semilinear PDE driven by a Bessel type generator$L^{\delta }$ with low dimension $0<\delta <1$. $L^{\delta }$ is a local operator, whose drift is thederivative of $x\mapsto \log (\mathrm{\mid }x\mathrm{\mid })$:in particular it is a Schwartz distribution, whichis not the derivative of a continuous function.The solutions are intended in a duality (‘‘weak’’) sensewith respect to state space$L^2({\mathbb{R}}_{+},d\mu ),$ $\mu$ being an invariant measure for the Bessel semigroup.