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If $X_t$ is a (continuous) diffusion process solving $$ dX_t = b(t,X_t)dt + \sigma(t,X_t)dW_t , $$ then its infinitesimal generator, denoted by $L$ is of the form by $$ L(f) = b(x) \cdot \nabla_{x} f(x) + \frac1{2} \big( \sigma(x) \sigma(x)^{\top} \big) : \nabla_{x} \nabla_{x} f(x). $$ Thus, $L$ is a second-order differential operator.

Question

If $X_t$ is a stochastic process (not a diffusion), which admits an infinitesimal generator $L$, is it possible for $L$ to be of order greater than $2$?

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    $\begingroup$ I suppose the answer is "no", but for a definite answer one needs to know precisely what do we mean by a "stochastic process" and "infinitesimal generator". If $X_t$ is supposed to be Markovian and the infinitesimal generator is to be a local operator in $\mathbb{R}^n$, then it is necessarily a second-order operator. On the other hand, in "rough" spaces (like fractals or percolation clusters) diffusions usually have higher-order operators as generators, although it is not clear whether one would still call them "differential operators". $\endgroup$ – Mateusz Kwaśnicki Jan 24 at 16:20
  • $\begingroup$ I was thinking of a rough setting, would you happen to have a reference for this case? $\endgroup$ – AIM_BLB Jan 24 at 16:22
  • $\begingroup$ I am not an expect here, but there is a number of survey works with "Diffusions on fractals" or similar phrases in their titles, these should be a good entry point into the literature. For example, by Barlow, Strichartz, Kumagai, Kumagai. Disclaimer: I just found some of these on the web, I did not even look into them. $\endgroup$ – Mateusz Kwaśnicki Jan 24 at 16:37
  • $\begingroup$ Alternatively, is there a notion of infinitesimal generator for non-markovian processes which can achieve this? $\endgroup$ – AIM_BLB Jan 24 at 16:39
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    $\begingroup$ I do not know, although I very much doubt. Probabilistic constructions typically lead to positivity-preserving operators, while higher-order operators generate semigroups that do not preserve positivity. That said, you may like the paper by Allouba and Zheng from Ann. Probab. 29(4) (2001), where they link a very weird non-Markovian process to the bi-Laplacian. Is this what you were looking for? $\endgroup$ – Mateusz Kwaśnicki Jan 24 at 16:46

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