I don't think what you've written is exactly true. At least some of the rental property in San Francisco is rent controlled (possibly a lot of it if you believe this site: http://www.sftu.org/rentcontrol.html). If a person/family lives in a rent controlled unit, rent increases aren't necessarily tied to the rental market, but rather are limited to a percentage of the CPI (1.9% increase for the current year - http://www.sfrb.org/index.aspx?page=1501). Wealthy neighbors don't have much to do with it in these cases.

If you look at the actual data, poor people are less likely to move out after gentrification than before.

Nice article that covers this: http://www.slate.com/articles/news_and_politics/politics/201...

This never happens in san francisco, the rents go up very little every year.. less than inflation, so in real terms your rent gets cheaper every year.