Kolmogorov's axioms for probabilities are used to justify formulating probability theory in terms of measure theory. Mathematically, the theory of measures that take negative or even complex values is well-developed. So, to the extent that probability theory is just measure theory, you can say a lot is known about negative probabilities.

But probability theory is not just measure theory; it adds its own distinctive ideas. And I'm not aware of significant extensions of these distinctive extra ideas to negative probabilities, even though not just Dirac but Feynman broached the idea of doing so.

As the above link notes, the Wigner distribution associated to a function $\psi \in L^2(\mathbb{R}^n)$ is an interesting example of something that conceptually resembles a probability distribution, but takes negative values.