Cosmic inflation is the accelerated expansion phase of the early universe. We believe in its existence because it can generate primordial density perturbations from the quantum vacuum fluctuation, which can grow into the current cosmological structures such as galaxies and clusters. In fact, this scenario is beautifully consistent with observations of the cosmic microwave background and the large-scale structure. To calculate the perturbation by inflation in detail, the stochastic approach is helpful. It treats the inflation dynamics as the random Brownian motion and enables non-linear analyses, numerical simulations, etc. It is formulated as the conversion from the Feynman path integral to the Wiener process. However, its discretisation is chosen ad hoc so that physically desired properties are satisfied and its mathematical justification is not known. We, therefore, understand it in a mathematical way called the non-standard analysis.

The non-standard analysis adds infinite and infinitesimal numbers to real numbers in a rigorous way and reformulates calculus. It is known that the non-standard analysis can formulate both the Feynman path integral and the Wiener process. Hence it is expected to be helpful also to formulate the stochastic approach to inflation.

The multiple zeta value (MZV) is a generalization of the special value of the Riemann zeta function. There exists a deep connection between MZVs and Feynman diagrams (integrals). Feynman diagram involves complex calculations of integrals, representing interactions of elementary particles in quantum field theory, and MZVs appear as fundamental tools in these calculations, allowing physicists to express the results of these diagrams in terms of a finite number of constants. The iterated integral expression of MZVs on the projective line minus three points is the key to this connection. Just as it was believed (now disproved) that every Feynman diagram from the simplified quantum field theory is written in terms of MZVs, much is still unknown. An ultimate goal would be to reveal all zeta values beyond MZVs that appear in all Feynman diagrams.