We investigate inflationary Higgs dynamics and constraints on the Standard Model parameters assuming the Higgs potential, computed to next-to-next leading order precision, is not significantly affected by new physics. For a high inflationary scale H ~ 1014 GeV suggested by BICEP2, we show that the Higgs is a light field subject to fluctuations which affect its dynamics in a stochastic way. Starting from its inflationary value the Higgs must be able to relax to the Standard Model vacuum well before the electroweak scale. We find that this is consistent with the high inflationary scale only if the top mass mt is significantly below the best fit value. The region within 2σ errors of the measured mt, the Higgs mass mh and the strong coupling αs and consistent with inflation covers approximately the interval mtlesssim171.8 GeV+0.538(mh-125.5 GeV) with 125.4 GeVlesssimmhlesssim126.3 GeV. If the low top mass region could be definitively ruled out, the observed high inflationary scale alone, if confirmed, would seem to imply new physics necessarily modifying the Standard Model Higgs potential below the inflationary scale.