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?



Probability that a stochastic flow is near $0$
$\newcommand\ep\epsilon$ $\newcommand\R{\mathbb R}$ $\newcommand\Si{\Sigma}$ 
Let 
$$X^x\_t:=xe^{-ct|x|}$$
for some real $c>0$ and allreal $t\ge0$ and $x\in\R^n$. Then $X^x\_0=x$ for all $x$ and 
your SDE holds with $\mu(t,x)=-c|x|xe^{-ct|x|}$ and $\Si(t,x)=0$. Moreover,
$$\int\_{\R^n}|X^x\_1|\,dx=\int\_{\R^n}|x|e^{-c|x|}\,dx<\ep,$$
as desired, if $c=c\_\ep$ is large enough. 




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If you insist on $\Si(t,x)\ne0$, you can clearly make $P(\int\_{\R^n}|X^x\_1|\,dx<\ep)$ arbitrarily close to $1$, by approximation. 


