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Could the phrase "let P^x denote the law of X given initial datum x ..." be better explained? What law are we talking about? Is this in the sense of the solution for an initial value problem? I'm assuming P^x is a probability measure or something - maybe it should be stated.
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Since we are only really relying on the values of f(x) inside a ball of radius, say, 2R, we could use a smooth cut-off function outside that set. It won't change any of the values where we actually evaluate f, but f will then be of compact support.
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f(x) = |x|^2 is not a compactly supported function, so I don't see how this example can work.
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