Let $g(x,v)$, $(x,v)\in [0,\infty)\times [0,\infty) $ be an unknown function for which we have knowledge of the following moments/constraints:
$$\int_{x=0}^{\infty}\text{d} x \int_{v=0}^{\infty} \text{d}v \, g(x,v) = 1$$ $$\rho(x) = \int_{v=0}^{\infty} \text{d} v \, g(x,v) \geq 0 $$ $$\frac{d \rho}{d x} \leq 0$$ $$g(x,v) = g(x,-v)$$ $$\int_{v=0}^{\infty} \text{d} v \, v g(x,v) = 0 , \forall x $$ $$\sigma^2(x) = \int_{v=0}^{\infty} \text{d} v \, v^2 g(x,v) \geq 0$$
In addition we may consider that $\sigma^2(x) < M$, where $M$ is some positive upper bound.
My question: Are these conditions sufficient for $g(x,v) \geq 0$ in $ [0,\infty)\times [0,\infty) $ ? And if yes, can we prove it? Any pointers/directions to where to seek a solution most appreciated (I have physics background).