# Weak continuity of law

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?

• 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 $\nu_{t_1}\ne\nu_{t_2}$? Jan 5, 2021 at 15:39
• A function defined on what space? Jan 5, 2021 at 15:44

## 1 Answer

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 $$t$$.

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