Stability of stochastic differential equations

Question

Consider an SDE of the form $$dX^\mu_t = a(t, X^\mu_t) dt + \sigma(t, X^\mu_t) dB_t$$ with initial condition $X^\mu_0 \sim \mu$, where $\mu$ is some measure on $\mathbb{R}^d$. I am searching for stability results of the following form: $$\mu_n \longrightarrow \mu \qquad \Longrightarrow \qquad Law(X^{\mu_n}_t) \longrightarrow Law(X^\mu_t) \,.$$ Here the convergence could be in the weak topology, for example. As far as I understand, supposing some well-posedness theory, the mapping $$(\mu, t) \longrightarrow Law(X^\mu_t)$$ should be well-defined, and hence my question should make sense.

Perhaps I don't know the right terminology but my search hasn't been successfull so far.

I would like to apply the Fokker-Planck/Kolmogorov forward equation in a setting where the initial distribution is a Dirac-delta. Instead of using some weak formulation it would be easiest for me to use an approximation result and work with a smooth distribution of intitial data. That's why I am searching for such a stability result.

Thank you!

Answer 1

The following might help.

Let $\mu, \nu$ be two probability measures on $\mathbb R^d$. For simplicity, let $a$ be Lipschitz with constant $L$ and $\sigma\equiv 1$. Then

$$|X^\mu(t)-X^\nu(t)|\leq |x_0^\mu-x_0^\nu|+\int_0^t L|X^\mu(s)-X^\nu(s)|ds.$$

Gronwall's inequality gives that $$|X^\mu(t)-X^\nu(t)|\leq |x_0^\mu-x_0^\nu|e^{tL}.$$

Wasserstein bounds follow.