Markov Processes with Identical Bridges
نویسنده
چکیده
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 ∈ E, the conditional distribution of (Xs)0≤s≤t, given X0 = x = Xt, coincides with the conditional distribution of (Ys)0≤s≤t, given Y0 = x = Yt, then the infinitesimal generators of X and Y are related by L f = ψ−1LX(ψf) − λf , where ψ is an eigenfunction of L with eigenvalue λ ∈ R. 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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