Convergence in law for a family of random generalized functions

Question

Let $\Phi_a$ be a family of random generalized functions all belonging to the same Sobolev space with negative exponent, say $H^{-k}$ for $k>0$. Is it possible to speak about convergence in law in $H^{-k}$ for such family? One natural notion would be the following: the family of measures $\mu_{a}$ induced by $\Phi_a$ converge to some limiting measure $\mu$: however this measure would be defined on the negative Sobolev space $H^{-k}$: so how to understand the corresponding convergence? It means just that for any continuous and bounded function $f$ on $H^{-k}$ we have $\int_{H^{-k}}f d\mu_a \to \int_{H^{-k}} f d \mu$ where $\mu$ is the limiting measure?

Answer 1

Yes. The convergence in law or distribution for random variables, in general, means the weak convergence of their laws or probability distributions. For this last notion you need to have a fixed topological space $X$ where all these measures $\mu_a$ and $\mu$ live (as Borel probability measures). The definition of convergence is that for all bounded continuous functions $F$ on $X$, $\int_X F(x)d\mu_a(x)$ must converge to $\int_X F(x)d\mu(x)$.

Now this definition is just the entry point into a vast and well-developed theory with lots of ready-to-use theorems. There is nothing special about $X=H^{-k}$ and it is treated the same as any other topological space. Note that the best situation with respect to the aforementioned theory is when $X$ is a Polish space (or some space which is not too far from that like the spaces of Schwartz distribution $\mathscr{S}'$ and $\mathscr{D}'$). The good news is that Sobolev spaces in $\mathbb{R}^n$, being separable Banach spaces, are therefore Polish. This gives you access to results about tightness like Prokhorov's theorem etc. For some advanced recent results specific to convergence of probability measures in Besov spaces (which include Sobolev as particular cases), see for example the article "A tightness criterion for random fields, with application to the Ising model" by Furlan and Mourrat in EJP 2017.