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We investigate the asymptotic behavior as time goes to infinity of Hawkesprocesses whose regression kernel has $L^1$ norm close to one and power lawtail of the form $x^{-(1+\alpha)}$, with $\alpha\in(0,1)$. We in particularprove that when $\alpha\in(1/2,1)$, after suitable rescaling, their lawconverges to that of a kind of integrated fractional Cox-Ingersoll-Rossprocess, with associated Hurst parameter $H=\alpha-1/2$. This result is incontrast to the case of a regression kernel with light tail, where a classicalBrownian CIR process is obtained at the limit. Interestingly, it shows thatpersistence properties in the point process can lead to an irregular behaviorof the limiting process. This theoretical result enables us to give anagent-based foundation to some recent findings about the rough nature ofvolatility in financial markets.

Rough fractional diffusions as scaling limits of nearly unstable heavy tailed Hawkes processes