Fix $\epsilon>0$ and let $(\Omega,\mathcal{F},\mathcal{F}_t\mathbb{P})$ be a stochastic base.  Is there a Markov diffusion process $X_t^x$ both satisfying an SDE of the form:
$$
d X_t = \mu(t,X_t)dt + \Sigma(t,X_t)dW_t, X_0^x
$$
and such that the associated stochastic flow $\phi_X:x\to X_1^x$ satisfies 
$$
\mathbb{P}\left(
\int_{x \in \mathbb{R}^n} |\phi_X(x)| dx < \epsilon
\right)=1?
$$

If not, are there known large-deviation-type estimates on 
$\mathbb{P}\left(\int_{x \in \mathbb{R}^n} |\phi_X(x)| dx<\epsilon \right)$ in terms of $\epsilon$?