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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?

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  • $\begingroup$ It is unclear what you want. (i) If $f$ is defined on $\mathbb{R}^n\times [0,\infty)$, how can $f$ map $\nu_s$ to anything, given that $\nu_s$ is not in $\mathbb{R}^n\times [0,\infty)$? (ii) If $s\le t_1<t_2$, how can $f$ map $\nu_s$ both to $\nu_{t_1}$ and $\nu_{t_2}$, given that usually you will have $\nu_{t_1}\ne\nu_{t_2}$? $\endgroup$ Jan 5, 2021 at 15:39
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    $\begingroup$ A function defined on what space? $\endgroup$ Jan 5, 2021 at 15:44

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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 $t$.

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  • $\begingroup$ Do you have a place where I can look this up and read more (esp. in the $L^1$ case)? $\endgroup$
    – ABIM
    Mar 22, 2021 at 19:13

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