Vibration energy harvesters (VEH) convert the energy of ambient random vibrations to electrical power in applications including autonomous powering of small scale devices. Theoretically, a VEH is represented by a system of two coupled differential equations, one each for the mechanical degree of freedom and the electrical degree of freedom. The interaction between noise and nonlinearity (by design) in a VEH is yet to be fully explored. This is of potential interest in the emerging concept of VEH using nanoelectromechanical systems (NEMS) since stochastic effects assume greater significance in the smaller dynamical scales. Allied with the recent focus on tristable harvesters, we investigate several aspects of the influence of random excitation on harvested power. It is shown that the unique effects of white noise excitation when compared with deterministic excitation include: (1) increase in harvested power by up to 14.28 percent (2) the advantages of asymmetrically designed potentials being achieved at much lower excitation levels and (3) the threshold values for inter-well oscillations being altered in favor of higher harvested power. Together, the results indicate that the advantages of tristable harvesters are enhanced due to white noise excitation; a feature of potential significance in NEMS based harvesters.