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[For the entire collection see Zbl 0728.00017.]
Let \(X=(\{x_ t(\omega)\in\mathbb{R}^ n,\;0\leq t\leq 1\},\Omega,{\mathcal F},\mathbb{P})\) be a Markov diffusion, \(k(x,y)\) be a strictly positive continuous probability density on \(\mathbb{R}^ n\times\mathbb{R}^ n\), and \[ \mathbb{Q}(A):=\int \mathbb{P}(A\mid\;x_ 0=x,\;x_ 1=y)k(x,y)dxdy,\qquad A\in{\mathcal F}. \] Then the process \(Y=(\{x_ t\},\Omega,{\mathcal F},\mathbb{Q})\) is called reciprocal diffusion governed by \(X\). On the other hand, suppose \(Y\) is a reciprocal diffusion governed by \(X\), then there can uniquely be defined a matrix pair \(\bar a(t,y)\) and \(c_ x(t,y)\), which are thought of as the local semimartingale characteristics of \(Y\). One of the main theorems of the paper shows how the diffusion matrix \(a(t,y)\) and the drift vector \(b(t,y)\) of the diffusion \(X\) are related to the semimartingale characteristics of \(Y\). This result is closely related to A. J. Krener’s expansion for \(\mathbb{P}(x_ t\in dy\mid\;x_{t-h}=x,\;x_{t+h}=z)\) [Stochastics 24, No. 4, 393-422 (1988; Zbl 0653.60048)]. The author also presents two probabilistic formulae that shed some light on the probabilistic interpretation of the local characteristics.