# Diffusions - global and local

Suppose $dX_t = \mu(X_t)dt + \sigma(X_t)dW_t$ is a diffusion. Is there a sense in which the dynamics are "dominated" locally by the diffusion term, and dominated globally by the drift term?

If $\mu$ and $\sigma$ are constants, then the law of the iterated logarithm says that the contribution from the diffusion term is slightly greater than $\sqrt{t}$, whereas the contribution from the drift term is linear.

On the other hand, over small timescales, small variations in the noise dominate any estimate one can make of the drift term.

Does a similar principle apply to more general diffusions?

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If $\mu$ and $\sigma$ are not too wild, then the local dynamics are determined by the diffusion. However, the large scale behavior can be affected by the diffusion.