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?