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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

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