# 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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## 1 Answer

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.

One example which may help is to consider a potential hill. A particle placed slightly to the left of the peak will tend to roll down to the left. A particle placed slightly to the right will tend to roll to the right. However, the strengths of those tendencies depend on the diffusion. If you have two potential wells with a hill between them, the proportion of the time spent in the deeper well depends on the diffusion, not just on the shape of the well.

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