The PQCG action has higher derivative terms (hence it is renormalisable) but because it is computing a probability and not an amplitude, it doesn't suffer from negative norm ghosts or tachyons (which is what plagues higher derivative theories of quantum gravity). 3/

Here is the path integral for Brownian motion straight out of Feynman and Hibbs. It has higher derivatives, but just describes a free particle being hit by random forces. On average the acceleration (q double dot) is zero, but there are contributions away from this. 4/