Celebratio Mathematica — Chung

Excursions, moderate Markov processes and probabilistic potential theory

Works connected to K. Murali Rao

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K. L. Chung and M. Rao: “Equilibrium and energy,” Probab. Math. Statist. 1 : 2 (1980), pp. 99–108. MR 626304 Zbl 0502.60060 article

In this paper it is shown that the equilibrium measure \( \nu \) for a compact \( K \) in potential theory can be related with a unique invariant measure \( \pi \) for a discrete time Markov process by the formula \[ \pi(dy) = \phi(y)\,\nu(dy) .\] The chain has the transition function \( L(x,A) \), where \( L \) is the last-exit kernel in [Chung 1973]. For a general non-symmetric potential density \( u \) the modified energy \[ I(\lambda) = \iint \lambda(dx)\,u(x,y)\,\phi(y)^{-1}(dy) \] and the Gauss quadratic \[ G(\lambda) = I(\lambda) - 2\lambda(K) \] are introduced. Then \( G \) is minimized by \( \pi \) among all signed measures \( \lambda \) on \( K \) of finite modified energy, provided \( I \) is positive. This includes the classical symmetric case of Newtonian and M. Riesz potentials as a special case. The modification corresponds to a time change for the underlying Markov process. The positivity of \( I \) is established for a class of signed measures associated with continuous additive functionals in the sense of Revuz.

K. L. Chung and K. M. Rao: “A new setting for potential theory, I,” Ann. Inst. Fourier (Grenoble) 30 : 3 (1980), pp. 167–198. MR 597022 Zbl 0424.31004 article

In Hunt’s theory of Markov processes certain duality assumptions are made to generalize well known classical potential theoretic results such as F. Riesz representation theorem, uniqueness, existence of equilibrium potential, etc. A standard treatment developed by several subsequent authors can be found in [Blumenthal and Getoor 1968]. In a different direction, it was shown in [Chung 1973] under simple analytic conditions that the equilibrium measure is inherently linked to the last exit distribution of the process. It thus appears feasible that this last result, namely on “equilibrium principle” may be made the starting point to which other major results are related.

In this paper we exploit the line of thought in [Chung 1973] to derive some of these major results under the same set of conditions as in [Chung 1973]. It turns out that these sets of conditions automatically imply the existence of a dual. However, this will not be proved here.

K. L. Chung, M. Liao, and K. M. Rao: “Duality under a new setting,” pp. 23–38 in Seminar on stochastic processes, 1983 (University of Florida, Gainesville, FL, 1983). Edited by E. Çinlar, K. L. Chung, and R. K. Getoor. Progress in Probability and Statistics 7. Birkhäuser (Boston, MA), 1984. MR 902410 Zbl 0558.60056 incollection

Erhan Çinlar	Related
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