Fix $\epsilon>0$ and let $(\Omega,F,F_t\mathbb{P})$ be a stochastic base.  Is there a (Markov) diffusion process $X_t$ satisfying an SDE of the form:
$$
d X_t = \mu(t,X_t)dt + \Sigma(t,X_t)dW_t, X_0^x
$$
such that the (random) function $f_X:x\to X_1^x$ satisfies 
$$
\mathbb{P}\left(
\int_{x \in \mathbb{R}^n} |f_X(x)| dx < \epsilon
\right)=1?
$$
If not, can we estimate the probability that this holds?