Let $\omega$ be the caloric measure on a domain bounded by a Weierstrass-type graph. We prove that $\omega$ is a quasi-Bernoulli measure. Therefore, the multifractal formalism isavailable for such a measure. We can then give necessary and sufficient conditions which ensure that $\omega$ is supported by a set of small dimension and prove that this is the generic case. These results improve and generalize a earlier work due to R. Kaufman and J.M. Wu.