Celebratio Mathematica

K. L. Chung and K. M. Rao: “Sur la théorie du potentiel avec la fonctionnelle de Feynman–Kac,” C. R. Acad. Sci. Paris Sér. A-B 290 : 13 (1980), pp. A629–A631. MR 572652 Zbl 0439.31005 article

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 and K. M. Rao: “Feynman–Kac functional and the Schrödinger equation,” pp. 1–29 in Seminar on stochastic processes, 1981 (Northwestern University, Evanston, IL, April 1981). Edited by E. Çinlar, K. L. Chung, and R. K. Getoor. Progress in Probability and Statistics 1. Birkhäuser (Boston, MA), 1981. MR 647779 Zbl 0492.60073 incollection

Erhan Çinlar	Related
K. Murali Rao	Related
Ronald K. Getoor	Related

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
Ronald K. Getoor	Related
K. Murali Rao	Related
Ming Liao	Related

K. L. Chung and K. M. Rao: “General gauge theorem for multiplicative functionals,” Trans. Amer. Math. Soc. 306 : 2 (April 1988), pp. 819–836. MR 933320 Zbl 0647.60083 article

We generalize our previous work on the gauge theorem and its various consequences and complements, initiated in [Chung and Rao 1981] and somewhat extended by subsequent investigations (see [Chung and Zhao]). The generalization here is two-fold. First, instead of the Brownian motion, the underlying process is now a fairly broad class of Markov processes, not necessarily having continuous paths. Second, instead of the Feynman–Kac functional, the exponential of a general class of additive functionals is treated. The case of Schrödinger operator \( \Delta/2+\nu \), where \( \nu \) is a suitable measure, is a simple special case. The most general operator, not necessarily a differential one, which may arise from our potential equations is briefly discussed toward the end of the paper. Concrete instances of applications in this case should be of great interest.

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Kai Lai Chung

Brownian motion and potential theory

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