7.6 The probabilistic core: Mitoma confinement
Let \(M : \mathcal{S}(\mathbb {R},\mathbb {R}) \to \mathbb {R}\) be nonnegative, even, subadditive and lower semicontinuous, with \(M(n^{-1}\varphi ) \to 0\) as \(n \to \infty \) for every fixed \(\varphi \). Then \(M\) is continuous at \(0\). The proof is a Baire-category argument on the Fréchet space \(\mathcal{S}\): the closed sets \(\{ \varphi \mid \forall m \ge k,\ M(m^{-1}\varphi ) \le \varepsilon \} \) cover the space, so one has nonempty interior, and subadditivity converts this into a neighborhood of \(0\) on which \(M\) is small.
Setting: a countable nonempty time set \(T\), probability spaces \((\Omega _i, P_i)_{i \in \iota }\), and measurable dual-valued processes \(Z_i : T \to \Omega _i \to \mathrm{SchDual}\), under hypothesis (H): for every test function \(\varphi \) and \(\varepsilon {\gt} 0\) there is \(a {\gt} 0\) with \(P_i\{ \omega \mid \exists t,\ a {\lt} |Z_i(t,\omega )\varphi |\} \le \varepsilon \) uniformly in \(i\). Then for every \(\varepsilon {\gt} 0\) there are \(q \in \mathbb {N}\) and \(\delta {\gt} 0\) such that \(\sup _i \int \sup _t \bigl\| 1 - e^{i\, Z_i(t,\omega )\varphi }\bigr\| \, dP_i(\omega ) \le \varepsilon + 2\, \| \varphi \| _q^2/\delta ^2\) for all \(\varphi \). The proof applies Xia’s lemma to the functional \(M(\varphi ) = \sup _i \int \sup _t \frac{|Z\varphi |}{1+|Z\varphi |}\, dP_i\) and converts the resulting Schwartz neighborhood into a Hermite–Sobolev ball via the two-sided domination.
Under the same measurability hypothesis and hypothesis (H): for every \(\varepsilon {\gt} 0\) there are \(q\) and \(\delta {\gt} 0\) such that for all \(C {\gt} 0\) and all \(i\), \(P_i\bigl\{ \omega \, \big|\, \exists t\, \exists N,\ C^2 {\lt} \sum _{j {\lt} N} \langle Z_i(t,\omega ), e^{q+1}_j\rangle ^2 \bigr\} \le \kappa \bigl(\varepsilon + \tfrac {2}{\delta ^2} \cdot \tfrac {S_{q,q+1}}{C^2}\bigr)\), where \(\kappa = \sqrt{e}/(\sqrt{e}-1)\) is the Badrikian constant and \(S_{q,q+1} = \sum _j \| e^{q+1}_j\| _q^2\) is the (finite) Hilbert–Schmidt constant. The proof averages the characteristic-functional bound over finite-dimensional Gaussian test functions \(\varphi _y = \sum _{j{\lt}N} y_j e^{q+1}_j\) with \(y \sim \mathcal{N}(0, C^{-2})^{\otimes N}\) and uses a Badrikian-type indicator estimate.
Under the measurability hypothesis and the per-test-function uniform sup-tightness hypothesis (H), for every \(\eta {\gt} 0\) there are \(q \in \mathbb {N}\) and \(B {\gt} 0\) such that the polar ball \(K = \mathrm{polarBall}\bigl(B \cdot \| \cdot \| _{q+1}\bigr)\) is compact in the pointwise dual, and for every \(i\), \(P_i\bigl\{ \omega \, \big|\, \exists t \in T,\ Z_i(t,\omega ) \notin K\bigr\} \le \eta \). That is, the processes stay in one fixed compact dual ball, uniformly over the countable time set and uniformly in \(i\), with probability at least \(1-\eta \). The proof chooses \(C\) large in the Gaussian-averaging bound and identifies bounded coefficient partial sums with polar-ball membership. This is the uniform dual-ball confinement at the heart of Mitoma’s tightness criterion for \(\mathcal{S}'\)-valued processes.