The key to extending the Heston model to a fractional stochastic volatility model is to consider the equation for the characteristic function
in Heston’s notation (Heston-original) but with
where solves the Heston’s Riccati equation. The difference between this solution and the Comte-Renault approach is that they fractionally integrate the volatility of Heston’s model while this approach produces directly a closed form solution as follows: solve for using Heston’s Riccati equation and then integrate by to obtain the solution of an exact PDE for long memory fractional stochastic volatility.
Some more details are here (fracsv)