Can sulfuric acid itself be electrolysed?
After some searching, I found this 1956 Thesis(Reference), which describes the production of peroxydisulfuric by electrolyzing concentrated sulfuric acid. The yield varied according to the concentration of the acid, temperature, current, and the nature of the electrode. In the Thesis, the chemist noted that discharged $\ce{HSO4-}$ united together to form peroxydisulfuric acid ($\ce{H2S2O8}$). It further stated that  oxidation of sulfate ions by anodic oxygen was responsible for the peroxydisulfuric acid formation. Also, hydroxyl ion at anode played an important role in the acid formation.

The reaction mechanism was confirmed by electrolyzing sulfuric acid and ammonium sulfate at various concentration at $\pu{16\!- \! 20 ^\circ C}$ using platinum electrode and the reaction occurred in accordance to the theory. It was also confirmed that amount of peroxidic oxygen increased with increase in ammonium sulfate. The specific gravity of sulfuric acid and the current density was varied and accordingly the highest yield of 78.4% was obtained. It was also noted that increasing the current density increases the yield up to a certain point as  peroxydisulfuric acid was found to convert into peroxymonosulfuric acid. At that moment, a small trace of hydrogen peroxide was noted.

So, the chemical species formed during electrolysis are $\ce{HSO4-}$, $\ce{SO4^2-}$, peroxide ion, protons, trace of hydrogen peroxide, and other side products.

Reference:

Sin-Chou Fan, “The electrolytic production of peroxydisulfuric acid using periodically reversed direct current and alternating current superimposed on direct current,” MS Thesis, Virginia Polytechnic Institute, 1956.