It is quite challenging to ascertain whether a certain multivariable function is a characteristic function when its corresponding measure is not trivial to be or not to be a probability measure on ℝ^d. In this article, we formulate multi-zeta function-based high-dimensional discrete probability distributions with infinitely many mass points on ℤ^d and ℤ^d-valued random walks given by those convolutions in terms of multiple zeta functions. In particular, a necessary and sufficient condition is provided for some polynomial finite Euler products to yield characteristic functions.

We introduce a continuation method for spatially discretized models, while conserving the size and shape of the cells and lattices. This proposed method is realized using shift operators and nonlocal operators of convolution types. Using this method and the shift operator, the nonlinear spatially discretized model on the uniform lattices can be systematically converted into a spatially continuous model. In addition, by convolution with suitable kernels, we mollify the shift operator and approximate the spatially discretized models using nonlocal evolution equations. We also show that this approximation is supported by the singular limit analysis. The continuous models designed using our method can successfully replicate the patterns corresponding to those of the original spatially discretized models obtained from the numerical simulations.