## The Annals of Probability

### General gauge and conditional gauge theorems

#### Abstract

General gauge and conditional gauge theorems are established for a large class of (not necessarily symmetric) strong Markov processes, including Brownian motions with singular drifts and symmetric stable processes. Furthermore, new classes of functions are introduced under which the general gauge and conditional gauge theorems hold. These classes are larger than the classical Kato class when the process is Brownian motion in a bounded $C^{1,1}$ domain.

#### Article information

Source
Ann. Probab. Volume 30, Number 3 (2002), 1313-1339.

Dates
First available in Project Euclid: 20 August 2002

http://projecteuclid.org/euclid.aop/1029867129

Digital Object Identifier
doi:10.1214/aop/1029867129

Mathematical Reviews number (MathSciNet)
MR1920109

Zentralblatt MATH identifier
1017.60086

#### Citation

Chen, Zhen-Qing; Song, Renming. General gauge and conditional gauge theorems. Ann. Probab. 30 (2002), no. 3, 1313--1339. doi:10.1214/aop/1029867129. http://projecteuclid.org/euclid.aop/1029867129.

#### References

• [1] AIKAWA, H. (1998). Norm estimate of Green operator, perturbation of Green's function and integrability of superharmonic functions. Math. Ann. 312 289-318.
• [2] ANCONA, A. (1997). First eigenvalues and comparison of Green's functions for elliptic operators on manifolds or domains. J. Anal. Math. 72 45-92.
• [3] BAÑUELOS, R. (1991). Intrinsic ultracontractivity and eigenfunction estimates for Schrödinger operators. J. Funct. Anal. 100 181-206.
• [4] BASS, R. and BURDZY, K. (1995). Conditioned Brownian motion in planar domains. Probab. Theory Related Fields 101 479-493.
• [5] BLUMENTHAL, R. M. and GETOOR, R. K. (1968). Markov Processes and Potential Theory. Academic, New York.
• [6] BLUMENTHAL, R. M., GETOOR, R. K. and RAY, D. B. (1961). On the distribution of first hits for the sy mmetric stable processes. Trans. Amer. Math. Soc. 99 540-554.
• [7] BOGDAN, K. and By CZKOWSKI, T. (1999). Potential theory for the -stable Schrödinger operator on bounded Lipschitz domains. Studia Math. 133 53-92.
• [8] CHEN, Z.-Q. and SONG, R. (1997). Intrinsic ultracontractivity and conditional gauge for sy mmetric stable processes. J. Funct. Anal. 150 204-239.
• [9] CHEN, Z.-Q. and SONG, R. (1998). Estimates on Green's functions and Poisson kernels of sy mmetric stable processes. Math. Ann. 312 465-601.
• [10] CHEN, Z.-Q. and SONG, R. (2000). Intrinsic ultracontractivity, conditional lifetimes and conditional gauge for sy mmetric stable processes on rough domains. Illinois J. Math. 44 138-160.
• [11] CHUNG, K. L. and RAO, M. (1988). General gauge theorem for multiplicative functionals. Trans. Amer. Math. Soc. 306 819-836.
• [12] CHUNG, K. L. and ZHAO, Z. (1995). From Brownian Motion to Schrödinger's Equation. Springer, Berlin.
• [13] CRANSTON, M., FABES, E. and ZHAO, Z. (1988). Conditional gauge and potential theory for the Schrödinger operator. Trans. Amer. Math. Soc. 307 174-194.
• [14] DOOB, J. L. (1984). Classical Potential Theory and Its Probabilistic Counterpart. Springer, New York.
• [15] FALKNER, N. (1987). Conditional Brownian motion in rapidly exhaustible domains. Ann. Probab. 15 1501-1514.
• [16] FITZSIMMONS, P. J. (1987). On the excursions of Markov processes in classical duality. Probab. Theory Related Fields 75 159-178.
• [17] FITZSIMMONS, P. J. and GETOOR, R. K. (1995). Occupation time distributions for Lévy bridges and excursions. Stochastic Process. Appl. 58 73-89.
• [18] FITZSIMMONS, P. J., PITMAN, J. and YOR, M. (1993). Markovian bridges: construction, Palm interpretation, and splicing. In Seminar on Stochastic Processes (E. Çinlar, K. L. Chung and M. J. Sharpe, eds.) 101-134. Birkhäuser, Boston.
• [19] GETOOR, R. K. (1980). Transience and recurrence of Markov processes. Séminaire de Probabilités XIV. Lecture Notes in Math. 784 397-409. Springer, New York.
• [20] GETOOR, R. K. (1999). An extended generator and Schrödinger equations. Elec. J. Probab. 4.
• [21] GETOOR, R. K. and GLOVER, J. (1984). Riesz decompositions in Markov process theory. Trans. Amer. Math. Soc. 285 107-132.
• [22] HE, S. W., WANG, J. G. and YAN, J. A. (1992). Semimartingale Theory and Stochastic Calculus. Science Press, Beijing.
• [23] KUNITA, H. and WATANABE, T. (1965). Markov processes and Martin boundaries. Illinois J. Math. 9 485-526.
• [24] MARTIN, R. S. (1941). Minimal positive harmonic functions. Trans. Amer. Math. Soc. 49 137- 172.
• [25] MURATA, M. (1997). Semismall perturbations in the Martin theory for elliptic equations. Israel J. Math. 102 29-60.
• [26] PINCHOVER, Y. (1989). Criticality and ground states for second order elliptic equations. J. Differential Equations 80 237-250.
• [27] STOLLMANN, P. and VOIGT, J. (1996). Perturbation of Dirichlet forms by measures. Potential Anal. 5 109-138.
• [28] STURM, K.-T. (1991). Gauge theorems for resolvents with application to Markov process. Probab. Theory Related Fields 89 387-406.
• [29] VOIGT, J. (1986). Absorption semigroups, their generators, and Schrödinger semigroups. J. Funct. Anal. 67 167-205.
• [30] ZHAO, Z. (1986). Green's function for Schrödinger operator and conditioned Fey nman-Kac gauge. J. Math. Anal. Appl. 116 309-334.
• [31] ZHAO, Z. (1991). A probabilistic principle and generalized Schrödinger perturbation. J. Funct. Anal. 101 162-176.
• [32] ZHAO, Z. (1992). Subcriticality and gaugeability of the Schrödinger operator. Trans. Amer. Math. Soc. 334 75-96.
• SEATTLE, WASHINGTON 98195 E-MAIL: zchen@math.washington.edu DEPARTMENT OF MATHEMATICS UNIVERSITY OF ILLINOIS
• URBANA, ILLINOIS 61801 E-MAIL: rsong@math.uiuc.edu