How to integrate the Heat Kernel?

Question

Let

$P(x, y, t) := (4 \pi t)^{-v/2}\exp\left(\frac{-|x-y|^2}{4t}\right)$, with $x, y \in \mathbb{R}^v$ and $t \in \mathbb{R}$,

denote the Heat Kernel.

QUESTION: How to show that:

$$Q(x, y, t) := \int^t_0 P(x, y, s)\,\mathrm ds$$

behaves like: $|x - y|^\gamma \exp\left(\frac{-|x-y|^2}{4t}\right)$ in the region:

$$A :=\{(x, y) \mid |x-y| \leq 4 \sqrt{t}\}$$

for some suitably chosen $\gamma$?