After the two historical descriptions by Einstein and Langevin of Brownian motion, the now well known generators acting on p-forms, are on one side the Witten Laplacian (Einstein) and on the other side Bismut's hypoelliptic Laplacian (Langevin). The accurate computation of exponentially small eigenvalues has many applications, in particular for the design of effective molecular dynamics algorithms. In the case of the Witten Laplacian, I will present the result obtained a few years ago with D. Le Peutrec and C. Viterbo, which makes the connection between the various exponential scales of small eigenvalues and the bar code of persistent homology. This provides a natural topological extension of the well known Arrhenius law in the scalar case, for general potential functions not assumed to be Morse. I will also present the more recent result obtained with X. Sang and F. White, which provides the same determination of the different spectral exponential scales in terms of the persistent homology bar code, in the double asymptotic regime of large friction and small temperature for Bismut's hypoelliptic Laplacian.