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Markov Processes with Identical Bridges
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P. J. Fitzsimmons, University of California, San Diego
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Abstract
Let X and Y be time-homogeneous Markov processes with common state space
E, and assume that the transition kernels of X and Y admit densities with
respect to suitable reference measures. We show that if there is a time t>0
such that, for each x in E, the conditional distribution of [X(s), 0
<= s
<=
t], given X(0) = x = X(t), coincides with the conditional distribution of
[Y(s), 0
<= s
<=
t], given Y(0) = x = Y(t), then the infinitesimal generators of X and Y are
related by LY(f)=p^{-1}LX(p*f)-lambda*f, where p is an eigenfunction
of LX with real eigenvalue lambda. Under an additional continuity
hypothesis, the same conclusion obtains assuming merely that X and Y share a
"bridge" law for one triple (x,t,y). Our work extends and clarifies a
recent result of I. Benjamini and S. Lee.
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Full text: PDF
Pages: 1-12
Published on: July 5, 1998
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