Sample path properties of the local times of strongly symmetric Markov processes via Gaussian processes.

*(English)*Zbl 0762.60068Regularity properties of the sample path of local times of a Markov process have been intensely studied, beginning with the celebrated theorem of Trotter on the joint continuity in the Brownian case. See in particular E. S. Boylan [Ill. J. Math. 8, 19-39 (1964; Zbl 0126.337)], R. K. Getoor and H. Kesten [Compositio Math. 24, 277- 303 (1972; Zbl 0293.60069)], and M. T. Barlow [Ann. Probab. 16, No. 4, 1389-1427 (1988; Zbl 0666.60072)]. D. Williams [Proc. Lond. Math. Soc., III. Ser. 28, 738-768 (1974; Zbl 0326.60093)] pointed out that the result of Trotter can be tackled using the Ray-Knight’s theorems, which describe the process in the space variable of the Brownian local times taken at certain random times, in terms of squares of Bessel processes. Squares of Bessel processes are particular examples of squares of Gaussian processes. In this direction, E. B. Dynkin [J. Funct. Anal. 55, 344-376 (1984; Zbl 0533.60061)] obtained a so-called isomorphism theorem, relating the local times of a symmetric Markov process with the square of an associated Gaussian process. However, this relation is not as explicit as in the Ray-Knight theorem, although the later can be derived from the isomorphism theorem.

The guideline of the paper under review is that known results on regularity of the sample path of a Gaussian process [see in particular M. Ledoux and M. Talagrand, Probability in Banach spaces (Springer, New York, 1991)] can be combined with Dynkin’s isomorphism theorem and applied to the study of local times of a symmetric Markov process. Specifically, consider a strongly symmetric Markov process \(X\) with state space \(S\), and let \(u^ 1(x,y)\) be the 1-potential density. When the function \(u^ 1\) is finite everywhere, \(X\) has local times \(L=(L^ y_ t, (t,y)\in R_ +\times S)\). Moreover, \(u^ 1(x,y)\) is positive definite, and there exists a zero-mean Gaussian process \(G=(G(y), y\in S)\) with covariance \(u^ 1(x,y)\). Perhaps the most striking result of the paper is the following necessary and sufficient condition for the existence of a jointly continuous version of the local times: \(L\) is a.s. continuous if and only if \(G\) is a.s. continuous.

Since a fundamental theorem of M. Talagrand [Acta Math. 159, No. 1- 2, 99-149 (1987; Zbl 0712.60044)] gives necessary and sufficient conditions for the continuity of a Gaussian process \(G\) in terms of the metric \(d(x,y)=E((G(x)-G(y)^ 2)^{1/2})\), one has also these conditions for the continuity of \(L\). Results in the same vein are obtained for the continuity (respectively, the boundedness) of \(L\) on \(R_ +\times K\), where \(K\subset S\) is a compact set, and for the continuity (respectively, the boundedness) of \(L\) at a given point. In the case when \(S\) is a locally compact Abelian group and \(X\) a symmetric Lévy process, the associated Gaussian process \(G\) is stationary, and the theorem of Dudley and Fernique is available. For \(S=R\), one recovers then a particular case of the characterization due to Barlow and Hawkes of Lévy processes having continuous local times (Barlow and Hawkes do not assume symmetry). Further developments in the direction of this paper were made by the authors [J. Theor. Probab. 5, 791-825 (1992; Zbl 0761.60035) and the paper reviewed below].

In conclusion, this paper provides an important link between two different areas of probability, Gaussian processes and symmetric Markov processes. It is very clear and nicely written, and easy to read even for a non-specialist of either field.

The guideline of the paper under review is that known results on regularity of the sample path of a Gaussian process [see in particular M. Ledoux and M. Talagrand, Probability in Banach spaces (Springer, New York, 1991)] can be combined with Dynkin’s isomorphism theorem and applied to the study of local times of a symmetric Markov process. Specifically, consider a strongly symmetric Markov process \(X\) with state space \(S\), and let \(u^ 1(x,y)\) be the 1-potential density. When the function \(u^ 1\) is finite everywhere, \(X\) has local times \(L=(L^ y_ t, (t,y)\in R_ +\times S)\). Moreover, \(u^ 1(x,y)\) is positive definite, and there exists a zero-mean Gaussian process \(G=(G(y), y\in S)\) with covariance \(u^ 1(x,y)\). Perhaps the most striking result of the paper is the following necessary and sufficient condition for the existence of a jointly continuous version of the local times: \(L\) is a.s. continuous if and only if \(G\) is a.s. continuous.

Since a fundamental theorem of M. Talagrand [Acta Math. 159, No. 1- 2, 99-149 (1987; Zbl 0712.60044)] gives necessary and sufficient conditions for the continuity of a Gaussian process \(G\) in terms of the metric \(d(x,y)=E((G(x)-G(y)^ 2)^{1/2})\), one has also these conditions for the continuity of \(L\). Results in the same vein are obtained for the continuity (respectively, the boundedness) of \(L\) on \(R_ +\times K\), where \(K\subset S\) is a compact set, and for the continuity (respectively, the boundedness) of \(L\) at a given point. In the case when \(S\) is a locally compact Abelian group and \(X\) a symmetric Lévy process, the associated Gaussian process \(G\) is stationary, and the theorem of Dudley and Fernique is available. For \(S=R\), one recovers then a particular case of the characterization due to Barlow and Hawkes of Lévy processes having continuous local times (Barlow and Hawkes do not assume symmetry). Further developments in the direction of this paper were made by the authors [J. Theor. Probab. 5, 791-825 (1992; Zbl 0761.60035) and the paper reviewed below].

In conclusion, this paper provides an important link between two different areas of probability, Gaussian processes and symmetric Markov processes. It is very clear and nicely written, and easy to read even for a non-specialist of either field.

Reviewer: J.Bertoin (Paris)

##### MSC:

60J55 | Local time and additive functionals |

60G15 | Gaussian processes |

60G17 | Sample path properties |