# Kernel of the adjoint of the infinitesimal generator of Levy SDE

Consider S.D.Es driven by a combination of Brownian and non-Brownian Levy noise (like say Gamma). Then we know that the flow of the density of the S.D.E variable is given by the adjoint of the infinitesimal generator (say $${\cal L}^*$$) of the S.D.E .

• How often is it true that the S.D.E variable's density is converging to the kernel of $${\cal L}^*$$?

• Are there examples of (a) being able to show when the kernel of this $${\cal L}^*$$ is unique and non-trivial and (b) being actually be able to compute this density in the kernel in such a case?

For comparison consider this paper (https://epubs.siam.org/doi/pdf/10.1137/0325042) where it was shown that for the basic Brownian motion driven SDE, the iterates converge to the Gibbs' measure which happens to be the kernel of the corresponding $${\cal L}^*$$. (Is the asymptotic measure being the kernel of $${\cal L}^*$$ a coincidence?)

• My question is essentially about asking what is the analogue of such a result for non-Brownian Levy noise.

For the Brownian case more general results of this type can be seen in, https://www.sciencedirect.com/science/article/pii/S0304414902001503