Is there any notion of sensitivity for probabilistic Boolean functions? Sensitivity is defined here. Denoting the neighbors of $x$ in the Boolean cube as $N(x)$, we define the sensitivity to be $s(f, x) = \sum_{y \in N(x)} I(f(x) \neq f(y))$, where $I$ is $1$ if the statement inside is true, $0$ otherwise. I'd imagine that you could do the same thing but instead of the indicator function use the probability and a lot of the results from that paper could still go through but in a probabilistic way.

I'm mostly just curious if anyone has looked at this before, as I think the notion I stated works just fine, but if anybody has an alternative notion or a problem with my candidate I'd be interested in that.
This should work fine inasmuch as if $f$ is now a random function, then you just get the expected sensitivity:
$$\mathbb E s(f, x) = \sum_{y \in N(x)} \mathbb E I(f(x) \neq f(y)) = \sum_{y \in N(x)} \Pr(f(x) \neq f(y)).$$