Help with Random Walk/Drunk

Basically I am organizing a performance around a random walk (currently the "drunk" object) which ends when the max of the object is reached. I am attracted to the indeterminacy of not knowing how long this will take, but I also need to insure that this 'maxing out' occurs at some point.

My thought is to have the "floor" or minimum value raise throughout the walk but I can't quite work it out - the drunk object doesn't offer a min, and to "set" a value still allows the output to fall below that value (something I don't want).

I'm writing because I feel this is possible with some manipulation of your externals/chance patchers, but I can't quite get there.

Is this something that anyone attempted to achieve? I've tried playing with the essl externals but no dice (pun intended).

Thanks.