Eternal Inflation and Swampland Conjectures

We study if eternal inflation is realized while satisfying the recently proposed string Swampland criteria concerning the range of scalar field excursion, $\mathrm{\mid }\mathrm{\Delta }\varphi \mathrm{\mid }<\mathcal{D}\cdot M_{\mathrm{P}}$, and the potential gradient, $\mathrm{\mid }\mathrm{\nabla }V\mathrm{\mid }>c\cdot V\mathrm{/}{M}_{\mathrm{P}}$, where $\mathcal{D}$ and $c$ are constants of order unity, and $M_{\mathrm{P}}$ is the reduced Planck mass. We find that only the eternal inflation of chaotic type is possible for $c \sim {\cal O}(0.01)$and$1/$\mathcal{D}$ \sim {\cal O}(0.01)$, and that the Hubble parameter during the eternal inflation is parametrically close to the Planck scale, and is in the range of $2 \pi c \lesssim H_{\rm inf}/M_{\rm P} < 1/\sqrt{3}$.