mu-Brownian Motion, Dualities, Diffusions, Transforms, and Reproducing Kernel Hilbert Spaces

Replacing the Lebesgue measure on an interval by a Stieltjes positive non-atomic measure, we study the corresponding counterpart of the Brownian motion. We introduce a new heat equation associated with the measure and make connections with stationary-increments Gaussian processes. We introduce a new transform analysis, and heat equation, associated with the measure, and make connections here too with stationary-increments and stationary Gaussian processes. In the main result of this paper (Theorem 7.2), we use white noise space analysis to derive a new heat equation associated with a (wide class of) stationary-increments Gaussian processes.