  This paper studies the infinite-dimensional Bayesian inference method with
Hadamard fractional total variation-Gaussian (HFTG) prior for solving inverse
problems. First, Hadamard fractional Sobolev space is established and proved to
be a separable Banach space under some mild conditions. Afterwards, the HFTG
prior is constructed in this separable fractional space, and the proposed novel
hybrid prior not only captures the texture details of the region and avoids
step effects, but also provides a complete theoretical analysis in the infinite
dimensional Bayesian inversion. Based on the HFTG prior, the well-posedness and
finite-dimensional approximation of the posterior measure of the Bayesian
inverse problem are given, and samples are extracted from the posterior
distribution using the standard pCN algorithm. Finally, numerical results under
different models indicate that the Bayesian inference method with HFTG prior is
effective and accurate.
