On Infinitesimal Generators and Feynman-Kac Integrals of Adelic Diffusion

For each prime $p$, a Vladimirov operator with a positive exponent specifies a $p$-adic diffusion equation and a measure on the Skorokhod space of $p$-adic paths. The product, $P$, of these measures with fixed exponent is a probability measure on the product of the $p$-adic path spaces. The adelic paths have full measure if and only if the sum, $\sigma$, of the diffusion constants is finite. Finiteness of $\sigma$ implies that there is an adelic Vladimirov operator, $\Delta_{\mathbb A}$, and an associated diffusion equation whose fundamental solution gives rise to the measure induced by $P$ on an adelic Skorokhod space. For a wide class of potentials, the dynamical semigroups associated to adelic Schr\"{o}dinger operators with free part $\Delta_{\mathbb A}$ have path integral representations.