Is there any way to replace the back button using an application?
This is possible if you're rooted. There's two applications that I know of which give you a little persistent overlay on the side of the screen, then pop out a full set of soft buttons when you press the overlay. One is SoftKeys, the other is Button Savior.

I haven't seen anything equivalent for non-rooted devices, but there is an app called SoftHome which adds a home button to your notification bar. It doesn't have a back button, however, just home.