Let $X$ be a smooth projective variety over a field $k$.
It is well-known that the sum of an ample divisor and an effective divisor is big (in fact this can be taken to be the definition of a big divisor). I'm looking for a weakening of this.
Is the sum of a big divisor and a pseudo-effective divisor itself a big divisor?
Recall that a divisor is called pseudo-effective if it lies in the closure of the cone of effective divisors.