The common linear energy harvesters have an inherent shortcoming in their operation concept because they operate efficiently only when the excitation frequency is very close to the fundamental frequency of the harvester. To extend the harvester's bandwidth, some recent solutions call for utilizing energy harvesters with stiffness-type nonlinearities. From a steady-state perspective, this hardening nonlinearity can extend the coupling between the excitation and the harvester to a wider range of frequencies. In this effort, we investigate the response of such harvesters to Gaussian White and Colored excitations. For Gaussian White excitations, we solve the Fokker-Plank-Kolmogorov equation for the exact joint probability density function of the response. We show that the expected value of the output power is not even a function of the non-linearity. As such, under White Gaussian excitations, nonlinearities in the stiffness do not provide any enhancement over the typical linear harvesters. Furthermore, we demonstrate that nonlinearities in the damping or inertia should be sought to enhance the output power. We also use the Van Kampen expansion to analyze the response to Colored excitations of different bandwidths and center frequencies. Again, we show that, regardless of the bandwidth or the center frequency, the expected value of the power is always less than that associated with a linear harvester.