Probabilistic approach to the Neumann problem
In this paper, we consider the Neumann boundary value problem of Schrödinger operator with measure potential μ. First, a martingale formulation of the Neumann problem and an analytic characterization of the martingale formulation are given. Then, by using the Dirichlet forms and Stochastic analysis we obtain an explicit formula for the unique weak solution of this problem in terms of reflecting Brownias motion and it's boundary local time.
KeywordsWeak Solution Local Time Explicit Formula Neumann Boundary Probabilistic Approach
Unable to display preview. Download preview PDF.
- Pei Hsu, Probabilistic Approach to the Neumann Problem.Comm. Pure Appl. Math.,38: 4(1985), 444–472.Google Scholar
- Ph. Blanchard, Zhiming Ma, Semigroup of Schrodinger Operators with Potentials Given by Radon Measure, BiBos Preprint, No. 262, 1987.Google Scholar
- M. Fukushima, Dirichlet Forms and Markov Processes, North-Holland Publ. Co. and Kodansha, 1980.Google Scholar
- Pei Hsu, Reflecting Brownian Motion, Boundary Local Time and the Neumenn Problem, Ph. D. Dissertations, Stanford University, 1984.Google Scholar
- Zhang Tusheng, The Characterization of Local Time ofd-dimensional Brownian Motion and Representation Theorems of Additive Functional,Acta Mathematica Sinica (in Chinese),32, 2 (1989), 161–173.Google Scholar