Let $\mathcal{P}\_2(\mathbb{R}^n)$ denote the set of all Borel probability measures on $\mathbb{R}^n$ with finite variance and *weak* topology. Let $X\_t$ be a strong solution to the SDE with initial conditions
$$
dX\_t = \mu(t,X\_t)dt + \sigma(t,X\_t) dW\_t, \mbox{ } X\_0=x
$$
for some Lipschitz-continuous functions $\mu$ and $\sigma$, and a Brownian motion $W\_t$. Denote its conditional law $\mathbb{P}(X\_t \in \cdot|X\_0=x)$.


**My Question:**
Is the map $(x,t)\mapsto \mathbb{P}(X\_t \in \cdot|X\_0=x)$ from $\mathbb{R}^n\times [0,\infty)$ to $\mathcal{P}\_2(\mathbb{R}^n)$ ever continuous?



Weak continuity of law
You can write $$X\_t-X\_t'=x-x'+\int\_0^t(\mu(s,X\_s)-\mu(s,X\_s'))ds+\int\_0^t(\sigma(s,X\_s)-\sigma(s,X\_s'))dW\_s $$
Then
$$ \mathbb{E}(\|X\_t-X\_t'\|^2) \leq 3 \left(|x-x'|^2+a^2t\int\_0^t \mathbb{E}(\|X\_s-X\_s'\|^2)ds + b^2\int\_0^t\mathbb{E}(\|X\_s-X\_s'\|^2) ds \right)$$ with $a,b$ the lipschitz constant of $\mu$ and $\sigma$. You can therefore use Gronwall and get that for all $t\leq T$ we have $$ \mathbb{E}(\|X\_t-X\_t'\|^2)\leq 3|x-x'|^2\exp(Ct) $$ for some $C>0$. In particular it is continue as a function on $x$ in $L^2$ and then for the weak topology.


Similarly, one can estimate $\mathbb{E}(\|X\_t-X\_{t+\delta t}\|^2)$ to have the continuity as a function on $t$.


