The Generators of a Gaussian Wave Associated with the Free Markov Field

Abstract

Suppose $\Phi = \{\phi_a; a \in A\}$ is a Gaussian random field. Let $\rho$ be a function on the parameter set $A$ with values in an open interval $I$. To every $t$ in $I$, there corresponds a subfield $\Phi_t = \{\phi_a; \rho(a) = t\}$ of the field $\Phi$. The family $\Phi_t, t \in I$, can be viewed as a Gaussian stochastic process. With a proper modification, this setup can be applied to generalized random fields for which the values at single points are not defined, in particular to the free field. In the case of a linear function $\rho$, the Gaussian process $\Phi_t$ plays a fundamental role in quantum field theory. It is a stationary Gaussian Markov process, where its Markov semigroup is given by the Feynman-Kac-Nelson formula. We prove that for a wide class of functions $\rho, \Phi_t$ is a nonhomogeneous Markov process and we evaluate the generators of this process.