Fix $\epsilon>0$ and let $(\Omega,\mathcal{F},\mathcal{F}_t,\mathbb{P})$ be a stochastic base, and let $f:\mathbb{R}^n\to \mathbb{R}$ be a continous function with $f(0)=0$. Is there a family of Markov diffusion processse $X_t^{x,\epsilon}$ satisfying an SDE of the form: $$ d X_t^{x,\epsilon} = \mu(t,X_t^{x,\epsilon},\epsilon)dt + \Sigma(t,X_t^{x,\epsilon},\epsilon)dW_t, X_0^{x,\epsilon} $$ and such that $ \mathbb{P}\left( \sup_{t \in [0,1]}|X_t^{x,\epsilon} - f(x)| < \epsilon \right) $ holds with high probability?

Note: Of course, here, we require that $\Sigma(t,x,\epsilon)>0$ to avoid trivial solutions.

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?

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 $\epsilon$?

Fix $\epsilon>0$ and let $(\Omega,\mathcal{F},\mathcal{F}_t,\mathbb{P})$ be a stochastic base, and let $f:\mathbb{R}^n\to \mathbb{R}$ be a continous function with $f(0)=0$. Is there a family of Markov diffusion processse $X_t^{x,\epsilon}$ satisfying an SDE of the form: $$ d X_t^{x,\epsilon} = \mu(t,X_t^{x,\epsilon},\epsilon)dt + \Sigma(t,X_t^{x,\epsilon},\epsilon)dW_t, X_0^{x,\epsilon} $$ and such that $ \mathbb{P}\left( \sup_{t \in [0,1]}|X_t^{x,\epsilon} - f(x)| < \epsilon \right) $ holds with high probability?

Note: Of course, here, we require that $\Sigma(t,x,\epsilon)>0$ to avoid trivial solutions.

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?

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 $\epsilon$?