Permafrost consists of soil and rocks that remain at 0 degrees C or below for at least two consecutive years. In mountains, permafrost ground ice acts like cement, stabilizing rock walls. Its degradation, following climate warming, may lead to slope instability in high mountains and damage to infrastructure, so knowledge about its evolution is essential for risk analysis. In pure solids, heat is transferred by conduction, but permafrost ground is also subject to non-conductive fluxes, and heat transfers are influenced by factors such as air temperature and snow cover, so a deterministic scheme cannot fully describe heat propagation. Current approaches to modelling use numerical models involving heat conduction schemes and energy balance models, requiring data on quantities such as relative humidity and radiation. We describe a stochastic treatment of the heat equation, which adapts to space-time changes in heat transfers driven by factors such as air temperature and snow cover, without requiring corresponding data, as part of a statistical model. The flexibility and performance of our approach are illustrated using data from two boreholes in the Swiss Alps, which show the strong influence of snow cover on ground temperature and the long-term degradation of permafrost produced by the 2003 heat wave.