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Stuart_Armstrong comments on A proposed inefficiency in the Bitcoin markets - Less Wrong
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Investments in Euros, Dollars and Yens are random walks on a log scale (because of the interest rates on offer in these currencies). Now, Bitcoin doesn't have any banks paying interest, as far as I know. But the market will still drive it towards random walks on a log scale, simply by people entering and leaving the market depending on how its expected value and risk compares with other commodities and investments. Random walks on log scales are the "natural" state of any investment.
Now people also hold bitcoins for non-investment purposes (as medium of exchange, as a political statement, etc...) But people also hold other goods for non-investment purposes (as a consumption good, for instance). So I don't see why bitcoin would differ from the usual financial rules.
Is this what you mean by "random walks on log scales are the 'natural' state of any investment": Most assets have fundamental reasons why they grow exponentially, and the assets which don't must therefore fall exponentially. Anything else going on?
Ok, we're at the very limits of my understanding, so don't assume that this is exactly correct, but...
Take the risk-free rate, either how it's standardly defined, or by just the size of the whole economy. The risk free rate is exponential, but that's an artefact of it being a "rate". You can have sub-exponential or super-exponential rates of growth of the whole economy, by varying the risk-free rate from year to year (or from moment to moment).
Then, in a well traded market, for reasons akin to what I mentioned above, every asset will be a random walk on the log scale, with the risk-free rate as the origin (ie if we continually adjust the values by the risk-free rate, we will get such a random walk).
Ok, that seems consistent with what I said.