Quantitative unique continuation and observability on an equidistributed set for the diffusion equation in <inline-formula><tex-math id="M1">$ \mathbb R^N $</tex-math></inline-formula>

In this paper, we obtain a quantitative estimate of unique continuation and an observability inequality from an equidistributed set for solutions of the diffusion equation in the whole space $ \mathbb R^N $. This kind of observability indicates that the total energy of solutions can be controlled by the energy localized in a measurable subset, which is equidistributed over the whole space. The proof of our results is based on an interesting reduction method [21,25], as well as the propagation of smallness for the gradient of solutions to elliptic equations [27].