Existence and functoriality of heat kernels on abstract Wiener groups

Let $\mathbf H\subset\mathbf G$ be an abstract Wiener group with Lie algebras $\mathbf h\subset\mathbf g$, and let $\nu_{t/N}^{\mathbf h\subset\mathbf g}$ denote the corresponding finite-dimensional approximating heat-kernel measures. For a map $P:(\mathbf H_1\subset\mathbf G_1)\to(\mathbf H_2\subset\mathbf G_2)$ of abstract Wiener groups, let $P_*$ denote pushforward. Abstract Wiener-group heat-kernel conjecture. For fixed $t\geq0$, the sequence

$$\exp_*(\nu_{t/N}^{\mathbf h\subset\mathbf g})^N$$

has a weak limit with respect to bounded continuous functions on $\mathbf G$. These limits, denoted $\nu_t^{\mathbf H\subset\mathbf G}$, form a convolution semigroup of inversion-invariant probability measures, and

$$P_*\nu_t^{\mathbf H_1\subset\mathbf G_1}=\nu_t^{\mathbf H_2\subset\mathbf G_2}$$

for every $t\geq0$. This is presented as the ideal existence result underlying the paper's infinite-dimensional heat-kernel construction.