ABSTRACT
We study a family of problems from measurable dynamics and their connection to the theory of positive definite functions. In particular, the aim of the present chapter is two-fold. One is an extension of the traditional setting for reproducing kernel Hilbert space (RKHS) theory. We also extend the traditional context of Aronszajn Aronszajn (1943, 1950) to better adapt it to a host of applications to probability theory, stochastic processes Alpay and Jorgensen (2012); Applebaum (2009); Jorgensen and Tian (2016); Mumford (2000), mathematical physics Haeseler et al. (2014); Konno et al. (2005); Osterwalder and Schrader (1973); Parussini et al. (2017); Rodgers et al. (2005); Tzeng and Wu (2006) and measurable dynamics, specifically reversible processes (see, e.g., Bezuglyi et al. (2014); Chang et al. (2015); Dutkay and Jorgensen (2011); Hersonsky (2012); Roblin (2011); Skopenkov (2013); Tosiek and Brzykcy (2013), and also Bishop and Peres (2017); Lubetzky and Peres (2016); Peres et al. (2016); Peres and Sousi (2016)). For applications to random processes, a kernel in the sense of Aronszajn will typically represent a covariance kernel.
