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I am studying a system subject to random noise, or a system driven by some noise, for example, heat flow or wave propagation perturbed by noise. I would like to know if there is a real system where the noise take effects slowly instead from the very beginning.

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Are you looking for a noise driven system where the noise has varying amplitude? Or maybe something else? It is difficult to tell from the question, a bit more detail may be appropriate. – tmac May 17 '12 at 7:11
Dear tmac, I am looking for a noise driven system where in the long run, the noise has fixed strength or amplitude. But in the beginning, the amplitude is small. Are there such physical scenarios? Thanks a lot. :-) – Anand May 18 '12 at 9:49

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It's not clear what you mean by "noise" but if you accept black-body radiation as electromagnetic noise then the average peak and total power output ramps up slowly for many systems including something as simple as a light bulb.

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