If the escalator is moving with speed v_1, and you are walking on the escalator with speed v_2 (that is, relative to the escalator you are moving with speed v_2), the actual speed your moving with will be v_1 + v_2.

So if you choose v_2 = – v_1, (that is, your relative speed to the escalator is exactly the opposite of the speed of the escalator; you’re walking the “wrong way”), someone who isn’t on the escalator will see you as standing still, even though you are actually walking.

It might seem odd, but how fast you can choose v_2 doesn’t depend on v_1 at all. It is just as fast as you can walk up or down an ordinary stair. (If v_1 is very fast, wind resistance would of course be a problem, but lets ignore that 🙂

Let’s say that v_2 is the fastest you can run on normal stairs. Then the fastest you run on an escalator, relative to the escalator, is also v_2 (ignoring wind resistance of course :).
If v_1 is very small compared to v_2 (the escalator is very slow), your actual speed will be v_1 + v_2. So even if you’re running very fast, the escalator still helps.

The difference in speed between running on the stairs and running on the escalator is v_1.

By the way, the picture at the top of this blog, http://mika.yukidoke.org/nikki/wp-content/themes/anthurium/images/banner.jpg
is very beautiful. Is it part of a bigger picture?