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Covariances

For positive measurable functions v and w, applying Kac's formula to tex2html_wrap_inline2808 , tex2html_wrap_inline3354 and tex2html_wrap_inline3356 and examining the result yields

  equation1882

where the terms in the decomposition can be understood using (16) and the occupation measure identity:

equation1888

These formulae are related to an energy form associated with v and w [35, 27, 64]. The tex2html_wrap_inline2992 covariance of tex2html_wrap_inline2808 and tex2html_wrap_inline3354 is

  equation1890

For each initial distribution tex2html_wrap_inline2820 , (24) is a symmetric bilinear non-negative definite function of pairs of non-negative functions chosen from tex2html_wrap_inline3370 . Similar remarks apply to the more general CAFs considered in Subsection 3.2.



Patrick Fitzsimmons
Wed May 17 08:50:36 PDT 2000