2000 character limit reached
Probabilistic Method to Fundamental gap problems on the sphere (2310.02808v2)
Published 4 Oct 2023 in math.PR and math.DG
Abstract: We provide a probabilistic proof of the fundamental gap estimate for Schr\"odinger operators in convex domains on the sphere, which extends the probabilistic proof of F. Gong, H. Li, and D. Luo for the Euclidean case. Our results further generalize the results achieved for the Laplacian by S. Seto, L. Wang, and G. Wei, as well as by C. He, G. Wei, and Qi S. Zhang. The essential ingredient in our analysis is the reflection coupling method on Riemannian manifolds.
- Proof of the fundamental gap conjecture. J. Amer. Math. Soc., 24(3):899–916, 2011.
- A note on first eigenvalue estimates by coupling methods in Kähler and quaternion Kähler manifolds. Electron. Commun. Probab., 27:Paper No. 12, 8, 2022.
- Application of coupling method to the first eigenvalue on manifold. Sci. China Ser. A, 37(1):1–14, 1994.
- General formula for lower bound of the first eigenvalue on riemannian manifolds. Science in China Series A: Mathematics, 40(4):384–394, 1997.
- M. Cranston. Gradient estimates on manifolds using coupling. J. Funct. Anal., 99(1):110–124, 1991.
- Fundamental gap estimate for convex domains on sphere—the case n=2𝑛2n=2italic_n = 2. Comm. Anal. Geom., 29(5):1095–1125, 2021.
- Rick Durrett. Probability—theory and examples, volume 49 of Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge, fifth edition, 2019.
- A probabilistic proof of the fundamental gap conjecture via the coupling by reflection. Potential Anal., 44(3):423–442, 2016.
- Fundamental gap of convex domains in the spheres. Amer. J. Math., 142(4):1161–1191, 2020.
- Elton P. Hsu. Stochastic analysis on manifolds, volume 38 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2002.
- Maximal coupling of Euclidean Brownian motions. Commun. Math. Stat., 1(1):93–104, 2013.
- Brownian motion and stochastic calculus, volume 113. Springer Science & Business Media, 1991.
- Wilfrid S. Kendall. Nonnegative Ricci curvature and the Brownian coupling property. Stochastics, 19(1-2):111–129, 1986.
- Coupling of multidimensional diffusions by reflection. Ann. Probab., 14(3):860–872, 1986.
- Bernt Ø ksendal. Stochastic differential equations. Universitext. Springer-Verlag, Berlin, sixth edition, 2003. An introduction with applications.
- Continuous martingales and Brownian motion, volume 293. Springer Science & Business Media, 2013.
- Sharp fundamental gap estimate on convex domains of sphere. J. Differential Geom., 112(2):347–389, 2019.
- Feng-Yu Wang. Gradient estimates and the first Neumann eigenvalue on manifolds with boundary. Stochastic Process. Appl., 115(9):1475–1486, 2005.
Paper Prompts
Sign up for free to create and run prompts on this paper using GPT-5.
Collections
Sign up for free to add this paper to one or more collections.
