  We define multifractional Hermite processes which generalize and extend both
multifractional Brownian motion and Hermite processes. It is done by
substituting the Hurst parameter in the definition of Hermite processes as a
multiple Wiener-It\^o integral by a Hurst function. Then, we study the
pointwise regularity of these processes, their local asymptotic self-similarity
and some fractal dimensions of their graph. Our results show that the
fundamental properties of multifractional Hermite processes are, as desired,
governed by the Hurst function. Complements are given in the second order
Wiener chaos, using facts from Malliavin calculus.
