Synthetic Research of Probability Theory

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Our purpose is to contribute to Japanese society of probability theory by the total sum of research results of our investigators, who are experts of each field, in order to develop and make use of the traversality that probability theory originally has.We sponsored or co-sponsored 4 international conferences (Probability and Number theory, Large Scale Interaction Systems, 2 conferences on Mathematical Finance), 26 domestic symposia, in which we financially supported 285 participants including 13 from overseas.We got many results. Among them, we here present 3 results obtained by the head investigator. (1) By formulating the Monte Carlo method as stochastic game (gambling), and applying Kolmogorov's random number theory, we revealed the essential problem of sampling in the Monte Carlo method (2) The probability of two monic polynomials over any finite fields to be coprime is computed by an extended ergodic theorem in the adelic completion of the ring of polynomials. (3) We considered a randomized digamma function, and investigated the central limit theorem for them, whose results can be regarded as the fluctuated potential caused by randomly located electric charges on the half line.