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April, 1988 The Generators of a Gaussian Wave Associated with the Free Markov Field
Wei-Shih Yang
Ann. Probab. 16(2): 752-763 (April, 1988). DOI: 10.1214/aop/1176991785


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.


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Wei-Shih Yang. "The Generators of a Gaussian Wave Associated with the Free Markov Field." Ann. Probab. 16 (2) 752 - 763, April, 1988.


Published: April, 1988
First available in Project Euclid: 19 April 2007

zbMATH: 0644.60042
MathSciNet: MR929076
Digital Object Identifier: 10.1214/aop/1176991785

Primary: 60G60
Secondary: 60G15

Keywords: Brownian motion , Feynman-Kac-Nelson formula , free Markov field , Gaussian random field , generators , Levy-Khintchine measure

Rights: Copyright © 1988 Institute of Mathematical Statistics


Vol.16 • No. 2 • April, 1988
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