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?