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Let $\mathcal{L}$ be the infinitesimal generator of a Markov semi-group. I am looking for references that study powers of $\mathcal{L}$; i.e. $\mathcal{L}^n$, for $n\in\mathbb{N}$.

For example, if the Markov semi-group generated by $\mathcal{L}$ is contractive, is the subgroup generated by $\mathcal{L}^2$ contractive to a higher order?

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  • $\begingroup$ I'm having difficulties to understand the question. In general, $\mathcal{L}^2$ does not generate a semigroup, as can be easily seen be considering the spectrum. Apart from this, could you please explain what you mean by "contractive to a higher order"? $\endgroup$
    – Jochen Glueck
    Commented Dec 16, 2023 at 11:04
  • $\begingroup$ @JochenGlueck- Could you give me a reference for the fact, or perhaps an example? I apologize if this question is too elementary; I don't work in this field. $\endgroup$
    – matilda
    Commented Dec 16, 2023 at 14:06
  • $\begingroup$ Let $\mathcal{L}$ denote the generator of the heat semigroup on $L^1(\mathbb{R})$. The spectrum of $\mathcal{L}^2$ is $[0,\infty)$, so $\mathcal{L}^2$ does not generate a $C_0$-semigroup. $\endgroup$
    – Jochen Glueck
    Commented Dec 16, 2023 at 16:30

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