Given a (standard) Brownian Motion $W_t$ if we do some sort of scaling, inversion or reversal then we also get a Brownian motion.

I have seen proofs but the proofs only seem to rely on showing covariance is the minimum of two given (arbitrary) times and the continuity.

I can't seem to find a theorem or lemma in my book that states this is all that needs to be checked rather than entirely check the 4 parts of the definition of Brownian motion.

Could someone either state or give reference (which I can find online) to this? That is much simpler than having to prove by the definition.


Durrett's Probability Theory and Examples gives this equivalent formulation on page 355 and proves it. There is in fact one more (very important!) property that needs to be checked, namely that the process is Gaussian (property (a') in Durrett). But if you are starting with a Brownian motion and doing some transformation to it, this is likely trivial to check.


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