Time-reversal and elliptic boundary value problems
Zhen-Qing Chen and Tusheng Zhang
Source: Ann. Probab. Volume 37, Number 3 (2009), 1008-1043.
Abstract
In this paper, we prove that there exists a unique, bounded continuous weak solution to the Dirichlet boundary value problem for a general class of second-order elliptic operators with singular coefficients, which does not necessarily have the maximum principle. Our method is probabilistic. The time reversal of symmetric Markov processes and the theory of Dirichlet forms play a crucial role in our approach.
Primary Subjects: 60J70, 60J57
Secondary Subjects: 35R05, 31C25i, 60H05, 60G46
Keywords: Diffusion; time-reversal; Girsanov transform; Feynman–Kac transform; multiplicative functional; partial differential equation; weak solution; boundary value problem; quadratic form; probabilistic representation
Full-text: Access denied (no subscription detected)
We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.
If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aop/1245434027
Digital Object Identifier: doi:10.1214/08-AOP427
Zentralblatt MATH identifier:
05587822
Mathematical Reviews number (MathSciNet):
MR2537548
References
[1] Ancona, A. (1997). First eigenvalues and comparison of Green’s functions for elliptic operators on manifolds or domains. J. Anal. Math. 72 45–92.
[2] Chen, Z.-Q. (2002). Gaugeability and conditional gaugeability. Trans. Amer. Math. Soc. 354 4639–4679 (electronic).
[3] Chen, Z.-Q., Fitzsimmons, P. J., Kuwae, K. and Zhang, T.-S. (2008). Stochastic calculus for symmetric Markov processes. Ann. Probab. 36 931–970.
[4] Chen, Z.-Q., Fitzsimmons, P. J., Kuwae, K. and Zhang, T.-S. (2008). Perturbation of symmetric Markov processes. Probab. Theory Related Fields 140 239–275.
[5] Chen, Z.-Q. and Song, R. (2002). General gauge and conditional gauge theorems. Ann. Probab. 30 1313–1339.
[6] Chen, Z.-Q., Williams, R. J. and Zhao, Z. (1999). On the existence of positive solutions for semilinear elliptic equations with singular lower order coefficients and Dirichlet boundary conditions. Math. Ann. 315 735–769.
[7] Chen, Z.-Q. and Zhang, T.-S. (2002). Girsanov and Feynman–Kac type transformations for symmetric Markov processes. Ann. Inst. H. Poincaré Probab. Statist. 38 475–505.
[8] Chen, Z. Q. and Zhao, Z. (1995). Diffusion processes and second order elliptic operators with singular coefficients for lower order terms. Math. Ann. 302 323–357.
[9] Chung, K. L. and Zhao, Z. X. (1995). From Brownian Motion to Schrödinger’s Equation. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 312. Springer, Berlin.
[10] Fitzsimmons, P. J. (1995). Even and odd continuous additive functionals. In Dirichlet Forms and Stochastic Processes (Beijing, 1993) 139–154. De Gruyter, Berlin.
[11] Fitzsimmons, P. J. and Kuwae, K. (2004). Nonsymmetric perturbations of symmetric Dirichlet forms. J. Funct. Anal. 208 140–162.
[12] Freidlin, M. (1985). Functional Integration and Partial Differential Equations. Annals of Mathematics Studies 109. Princeton Univ. Press, Princeton, NJ.
[13] Fukushima, M. (1995). On a decomposition of additive functionals in the strict sense for a symmetric Markov process. In Dirichlet Forms and Stochastic Processes (Beijing, 1993) 155–169. De Gruyter, Berlin.
[14] Fukushima, M., Ōshima, Y. and Takeda, M. (1994). Dirichlet Forms and Symmetric Markov Processes. De Gruyter Studies in Mathematics 19. De Gruyter, Berlin.
[15] Gerhard, W. D. (1992). The probabilistic solution of the Dirichlet problem for ½Δ+〈a, ∇〉+b with singular coefficients. J. Theoret. Probab. 5 503–520.
[16] Gilbarg, D. and Trudinger, N. S. (1983). Elliptic Partial Differential Equations of Second Order, 2nd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 224. Springer, Berlin.
[17] Glover, J., Rao, M., Šikić, H. and Song, R. (1994). Quadratic forms corresponding to the generalized Schrödinger semigroups. J. Funct. Anal. 125 358–378.
[18] Kakutani, S. (1944). Two-dimensional Brownian motion and harmonic functions. Proc. Imp. Acad. Tokyo 20 706–714.
[19] Kunita, H. (1970). Sub-Markov semi-groups in Banach lattices. In Proc. Internat. Conf. on Functional Analysis and Related Topics (Tokyo, 1969) 332–343. Univ. Tokyo Press, Tokyo.
[20] Lunt, J., Lyons, T. J. and Zhang, T. S. (1998). Integrability of functionals of Dirichlet processes, probabilistic representations of semigroups and estimates of heat kernels. J. Funct. Anal. 153 320–342.
[21] Ma, Z. M. and Röckner, M. (1992). Introduction to the Theory of (nonsymmetric) Dirichlet Forms. Springer, Berlin.
[22] Meyers, N. G. (1963). An Lp-estimate for the gradient of solutions of second order elliptic divergence equations. Ann. Scuola Norm. Sup. Pisa (3) 17 189–206.
[23] Nakao, S. (1985). Stochastic calculus for continuous additive functionals of zero energy. Z. Wahrsch. Verw. Gebiete 68 557–578.
[24] Sharpe, M. (1988). General Theory of Markov Processes. Pure and Applied Mathematics 133. Academic Press, Boston.
[25] Stroock, D. W. (1988). Diffusion semigroups corresponding to uniformly elliptic divergence form operators. In Séminaire de Probabilités, XXII. Lecture Notes in Math. 1321 316–347. Springer, Berlin.
[26] Trudinger, N. S. (1973). Linear elliptic operators with measurable coefficients. Ann. Scuola Norm. Sup. Pisa (3) 27 265–308.
The Annals of Probability
