Intermittency for the stochastic heat and wave equations with generalized fractional noise

We are looking at the stochastic heat and wave equations with different types of fractional noise. We are interested in the intermittency property and Lyapunov exponent for the solution. First we look at the equation driven by the Dobric-Ojeda noise and we show that the Lyapunov exponent matches that of the equation driven by standard fractional noise as obtained by Hu, Huang, Nualart, Tindel (2015) and Balan, Conus (2016). In the second part, we introduce a generalized fractional noise that includes both standard fractional noise and Dobric-Ojeda noise. It shows that, in this specific situation, the correlation structure of the noise does not change the Lyapunov exponent. We conjecture that this result would hold more generally