As always, you will need to benchmark for your application / environment. And unless you have already profiled and idetified this as a performance bottleneck, it's probably not going to matter ("premature optimization is the root of all evil").
Having said that:
a.equals(b) is really fast for Strings. It's probably one of the most tightly optimized pieces of code in the Java platform. I'd be very surprised if you can find any faster way of comparing two arbitrary strings.
There are special cases where you can cheat and use (a==b) safely, e.g. if you know that both Strings are interned (and therefore value identity implies object identity). In that case it may be slightly faster than a.equals(b) - but again this depends on compiler/JVM implementation. And it's very easy to shoot yourself in the foot if you don't know what you are doing.....