We introduce a new family of processes that includes the long memory (LM) (power law) in the volatility correlation. This is achieved by measuring the historical volatilities on a set of increasing time horizons and by computing the resulting effective volatility by a sum with power law weights. The processes have two parameters (linear processes) or four parameters (affine process). In the limit where only one component is included, the processes are equivalent to GARCH(1, 1) and I-GARCH(1). Volatility forecast is discussed in the context of processes with quadratic equations, in particulars as a means to estimate process parameters. Using hourly data, the empirical properties of the new processes are compares to existing processes (GARCH, I-GARCH, FIGARCH, ...), in particular log-likehood estimates and volatility forecast errors. This study covers time horizons ranging from 1 h to 1 month. We also study the variation of the estimated parameters with respect to changing sample by introducting a natural coordinate invariant distance. The LM processes show a small but systematic quantitative improvement with respect to the standard GARCH(1, 1) process. Yet the main advantage of the new LM processes is that they give a good description of the empirical data from 1 h to 1 monthwith the same parameters. Their other advantages are that they are efficient to evaluate numerically, that they behave well with respest to the cut-off (i.e. the largest time horizon included in the process) and that they can be extended along several directions.