Abstract
AbstractWe discuss Neumann problems for self-adjoint Laplacians on (possibly infinite) graphs. Under the assumption that the heat semigroup is ultracontractive we discuss the unique solvability for non-empty subgraphs with respect to the vertex boundary and provide analytic and probabilistic representations for Neumann solutions. A second result deals with Neumann problems on canonically compactifiable graphs with respect to the Royden boundary and provides conditions for unique solvability and analytic and probabilistic representations.
Funder
Deutsche Forschungsgemeinschaft
Publisher
Springer Science and Business Media LLC
Subject
General Mathematics,Theoretical Computer Science,Analysis
Reference40 articles.
1. Probability and its Applications.;RF Bass,1995
2. Blumenthal, R.M., Getoor, R.K.: Markov Processes and Potential Theory. Dover Publications, New York (1968)
3. Bass, R.F., Hsu, P.: Some potential theory for reflecting Brownian motion in Hölder and Lipschitz domains. Ann. Probab. 19(2), 486–508 (1991)
4. Bensoussan, A., Menaldi, J.-L.: Difference equations on weighted graphs. J. Convex Anal. 12(1), 13–44 (2005)
5. Benchérif-Madani, A., Pardoux, É.: A probabilistic formula for a Poisson equation with Neumann boundary condition. Stoch. Anal. Appl. 27(4), 739–746 (2009)