Construction of Interaction Measures on the Space of Distributions over the Field of $p$-Adic Numbers

We overview the construction of non-Gaussian measures on the space $\mathcal D'(\mathbb Q_p^n)$, $n\le 4$, of Bruhat–Schwartz distributions over the field of $p$-adic numbers, corresponding to finite volume polynomial interactions in a $p$-adic analogue of the Euclidean quantum field theory. Our choice of the free measure is the Gaussian measure corresponding to an elliptic pseudodifferential operator over $\mathbb Q_p^n$. Analogues of the Euclidean $P(\varphi)$-theories with free and half-Dirichlet boundary conditions are considered.