A probabilistic representation for heat flow of harmonic map on manifolds with time-dependent Riemannian metric

Statistics and Probability Letters - Tập 177 - Trang 109165 - 2021
Xin Chen1, Wenjie Ye1
1School of Mathematical Sciences, Shanghai Jiaotong University, Shanghai 200240, China

Tài liệu tham khảo

Arnaudon, 2008, Brownian motion with respect to a metric depending on time: definition, existence and applications to Ricci flow, C. R. Math. Acad. Sci. Paris, 346, 773, 10.1016/j.crma.2008.05.004 Bismut, 1976, Linear quadratic optimal stochastic control with random coefficients, SIAM J. Control Optim., 14, 419, 10.1137/0314028 Chavel, 2006, vol. 98 Chen, 2020 Chen, 2021 Cheng, 2017, Diffusion semigroup on manifolds with time-dependent metrics, Forum Math., 29, 775, 10.1515/forum-2015-0049 Coulibaly, 2011, Brownian motion with respect to time-changing Riemannian metrics, applications to Ricci flow, Ann. Inst. Henri Poincaré Probab. Stat., 47, 515 do Carmo, 1992 Eells, 1964, Harmonic mappings of Riemannian manifolds, Amer. J. Math., 86, 109, 10.2307/2373037 Elworthy, 1999, vol. 1720 Elworthy, 2007, Itô maps and analysis on path spaces, Math. Z., 257, 643, 10.1007/s00209-007-0145-0 Funaki, 1992, A stochastic partial differential equation with values in a manifold, J. Funct. Anal., 109, 257, 10.1016/0022-1236(92)90019-F Guo, 2014, A stochastic approach to the harmonic map heat flow on manifolds with time-dependent Riemannian metric, Stochastic Process. Appl., 124, 3535, 10.1016/j.spa.2014.06.004 Hsu, 2002, vol. 38 Kendall, 1990, Probability, convexity, and harmonic maps with small image I: Uniqueness and fine existence, Proc. Lond. Math. Soc., 3, 371, 10.1112/plms/s3-61.2.371 Kupper, 2019, Multidimensional Markovian FBSDEs with super-quadratic growth, Stochastic Process. Appl., 129, 902, 10.1016/j.spa.2018.03.024 Ladyženskaja, 1968, vol. 23 Li, 1994, Strong p-completeness of stochastic differential equations and the existence of smooth flows on noncompact manifolds, Probab. Theory Related Fields, 100, 485, 10.1007/BF01268991 Li, 2014, Eigenvalues and entropies under the harmonic-Ricci flow, Pacific J. Math., 267, 141, 10.2140/pjm.2014.267.141 Lin, 2008 Ma, 1994, Solving forward-backward stochastic differential equations explicitly—A four step scheme, Probab. Theory Related Fields, 98, 339, 10.1007/BF01192258 Ma, 2012, On non-Markovian forward-backward SDEs and backward stochastic PDEs, Stochastic Process. Appl., 122, 3980, 10.1016/j.spa.2012.08.002 Mantegazza, 2011, vol. 290 Müller, 2012, Ricci flow coupled with harmonic map flow, Ann. Sci. Éc. Norm. Supér., 4, 101, 10.24033/asens.2161 Pardoux, 1990, Adapted solution of a backward stochastic differential equation, Systems Control Lett., 14, 55, 10.1016/0167-6911(90)90082-6 Pardoux, 1992, Backward stochastic differential equations and quasilinear parabolic partial differential equations, vol. 176, 200 Peng, 1991, Probabilistic interpretation for systems of quasilinear parabolic partial differential equations, Stoch. Stoch. Rep., 37, 61, 10.1080/17442509108833727 Röckner, 2020, Stochastic heat equations with values in a manifold via Dirichlet forms, SIAM J. Math. Anal., 52, 2237, 10.1137/18M1211076 Thalmaier, 1996, Brownian motion and the formation of singularities in the heat flow for harmonic maps, Probab. Theory Related Fields, 105, 335, 10.1007/BF01192212 Wu, 2014, Probabilistic interpretation for a system of quasilinear parabolic partial differential equation combined with algebra equations, Stochastic Process. Appl., 124, 3921, 10.1016/j.spa.2014.07.013