Probability that a stochastic flow is near $0$

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?

$$\newcommand\ep\epsilon$$ $$\newcommand\R{\mathbb R}$$ $$\newcommand\Si{\Sigma}$$ 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.
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.
• But I'm a bit confused. So in general, for a typical function we would take $X_t^x = x(e^{-ct|x|} +f(x))$?