  In this paper, we establish several Liouville type theorems for entire
solutions to fractional parabolic equations. We first obtain the key
ingredients needed in the proof of Liouville theorems, such as narrow region
principles and maximum principles for antisymmetric functions in unbounded
domains, in which we remarkably weaken the usual decay condition $u \to 0$ at
infinity with respect to the spacial variables to a polynomial growth on $u$ by
constructing auxiliary functions.Then we derive monotonicity for the solutions
in a half space $\mathbb{R}_+^n \times \mathbb{R}$ and obtain some new
connections between the nonexistence of solutions in a half space
$\mathbb{R}_+^n \times \mathbb{R}$ and in the whole space $\mathbb{R}^{n-1}
\times \mathbb{R}$ and therefore prove the corresponding Liouville type
theorems.
  To overcome the difficulty caused by the non-locality of the fractional
Laplacian, we introduce several new ideas which will become useful tools in
investigating qualitative properties of solutions for a variety of non-local
parabolic problems.
