Recently, in [Preprint (2006)], we extended the concept of intrinsic ultracontractivity to nonsymmetric semigroups. In this paper, we study the intrinsic ultracontractivity of nonsymmetric diffusions with measure-valued drifts and measure-valued potentials in bounded domains. Our process Y is a diffusion process whose generator can be formally written as L+μ⋅∇−ν with Dirichlet boundary conditions, where L is a uniformly elliptic second-order differential operator and μ=(μ1, …, μd) is such that each component μi, i=1, …, d, is a signed measure belonging to the Kato class Kd,1 and ν is a (nonnegative) measure belonging to the Kato class Kd,2. We show that scale-invariant parabolic and elliptic Harnack inequalities are valid for Y.
In this paper, we prove the parabolic boundary Harnack principle and the intrinsic ultracontractivity for the killed diffusion YD with measure-valued drift and potential when D is one of the following types of bounded domains: twisted Hölder domains of order α∈(1/3, 1], uniformly Hölder domains of order α∈(0, 2) and domains which can be locally represented as the region above the graph of a function. This extends the results in [J. Funct. Anal. 100 (1991) 181–206] and [Probab. Theory Related Fields 91 (1992) 405–443]. As a consequence of the intrinsic ultracontractivity, we get that the supremum of the expected conditional lifetimes of YD is finite.
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References
Bañuelos, R. (1991). Intrinsic ultracontractivity and eigenfunction estimates for Schrödinger operators. J. Funct. Anal. 100 181--206.
Bass, R. F. and Burdzy, K. (1992). Lifetimes of conditioned diffusions. Probab. Theory Related Fields 91 405--443.
Bass, R. F. and Chen, Z.-Q. (2003). Brownian motion with singular drift. Ann. Probab. 31 791--817.
Blumenthal, R. M. and Getoor, R. K. (1968). Markov Processes and Potential Theory. Academic Press.
Chen, Z.-Q. (2002). Gaugeability and conditional gaugeability. Trans. Amer. Math. Soc. 354 4639--4679.
Chen, Z.-Q. and Kumagai, T. (2003). Heat kernel estimates for stable-like processes on $d$-sets. Stochastic Process. Appl. 108 27--62.
Chung, K. L. (1986). Doubly--Feller process with multiplicative functional. In Seminar on Stochastic Processes, 1985 63--78. Birkhaüser, Boston.
Chung, K. L. (1984). The lifetime of conditional Brownian motion in the plane. Ann. Inst. H. Poincaré Probab. Statist. 20 349--351.
Chung, K. L. and Rao, K. M. (1980). A new setting for potential theory. Ann Inst. Fourier 30 167--198.
Chung, K. L. and Walsh, J. B. (2005). Markov Processes, Brownian Motion, and Time Symmetry. Springer, New York.
Davies, E. B. and Simon, B. (1984). Ultracontractivity and the heat kernel for Schrödinger operators and Dirichlet Laplacians. J. Funct. Anal. 59 335--395.
Davis, B. (1991). Intrinsic ultracontractivity and the Dirichlet Laplacian. J. Funct. Anal. 100 163--180.
Fabes, E. B. and Stroock, D. W. (1986). A new proof of Moser's parabolic Harnack inequality using the old ideas of Nash. Arch. Rational Mech. Anal. 96 327--338.
Kim, P. (2006). Relative Fatou's theorem for $(-\Delta)^\alpha/2$-harmonic function in $\kappa$-fat open set. J. Funct. Anal. 234 70--105.
Kim, P. and Song, R. (2006). Two-sided estimates on the density of Brownian motion with singular drift. Illinois J. Math. 50 635--688.
Kim, P. and Song, R. (2007). Boundary Harnack principle for Brownian motions with measure-valued drifts in bounded Lipschitz domains. Math. Ann. 339 135--174.
Kim, P. and Song, R. (2007). Estimates on Green functions and the Schrödinger-type equations for non-symmetric diffusions with measure-valued drifts. J. Math. Anal. Appl. 332 57--80.
Kim, P. and Song, R. (2006). Intrinsic ultracontractivity of non-symmetric diffusion semigroups in bounded domains. Tohoku Math. J. To appear.
Kim, P. and Song, R. (2008). On dual processes of non-symmetric diffusions with measure-valued drifts. Stochastic Process. Appl. 118 790--817.
Liao, M. (1984). Riesz representation and duality of Markov processes. Ph.D. dissertation, Dept. Mathematics, Stanford Univ.
Mathematical Reviews (MathSciNet):
MR889495
Liao, M. (1985). Riesz representation and duality of Markov processes. Lecture Notes in Math. 1123 366--396. Springer, Berlin.
Liu, L. Q. and Zhang, Y. P. (1990). Representation of conditional Markov processes. J. Math. (Wuhan) 10 1--12.
Pop-Stojanović, Z. R. (1988). Continuity of excessive harmonic functions for certain diffusions. Proc. Amer. Math. Soc. 103 607--611.
Pop-Stojanović, Z. R. (1989). Excessiveness of harmonic functions for certain diffusions. J. Theoret. Probab. 2 503--508.
Shur, M. G. (1977). Dual Markov processes. Teor. Verojatnost. i Primenen. 22 264--278. [English translation: Theory Probab. Appl. 22 (1977) 257--270.]
Schaefer, H. H. (1974). Banach Lattices and Positive Operators. Springer, New York.
Zhang, Q. S. (1996). A Harnack inequality for the equation $\nabla(a\nabla u)+b\nabla u=0$, when $\vert b\vert\in K\sbn+1$. Manuscripta Math. 89 61--77.
