Abstract
Our domain space will be a measurable space(M,M)with a given Markov semi-group
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References
W. Ballmann (1995), Lectures on spaces of nonpositive curvature. DMV Seminar Band 25, Birkhäuser.
H. Bauer (1996), Probability theory. de Gruyter Studies in Mathematics, 23, Walter de Gruyter & Co., Berlin.
A. Bendikov (1995), Potential theory on infinite-dimensional Abelian groups. de Gruyter Studies in Mathematics, 21., Walter de Gruyter,Berlin.
A. Bendikov, L. Saloff-Coste (2001), On the absolute continuity of Gaussian measures on locally compact groups. J. Theor. Probab. 14, No.3, 887-898.
J. Eells, B. Fuglede (2001), Harmonic maps between Riemannian polyhedra. Cam- bridge Tracts in Mathematics, 142., Cambridge University Press, Cambridge.
M. Emery (1989), Stochastic calculus in manifolds. Universitext. Springer, Berlin.
A. Es-Sahib, H. Heinich (1999), Barycentre canonique pour un espace métrique à courbure négative. Séminaire de Probabilités, XXXIII, 355-370, Lecture Notes in Math., 1709, Springer, Berlin.
S.N. Ethier, T.G. Kurtz (1986), Markov processes. Characterization and convergence. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley, New York.
C. Fefferman, D.H. Phong (1983), Subelliptic eigenvalue problems. Conference on harmonic analysis, Chicago, Wadsworth Math. Ser., 590-606.
A. Grigoryan (1999), Estimates of heat kernels on Riemannian manifolds. Spectral theory and geometry, 140-225, London Math. Soc. Lecture Note Ser., 273, Cambridge Univ. Press.
N. Jacob (1996), Pseudo-differential operators and Markov processes. Mathematical Research, 94, Akademie Verlag, Berlin.
N. Jacob (2001), Pseudo differential operators and Markov processes. Vol. I. Fourier analysis and semigroups. Imperial College Press, London.
D.S. Jerison, A. Sanchez-Calle (1986), Estimates for the heat kernel for a sum of squares of vector fields. Indiana Univ. Math. J. 35 no. 4, 835-854.
J. Jost (1994), Equilibrium maps between metric spaces. Calc. Var. Partial Differential Equations 2, 2, 173-204.
J. Jost (1997), Nonpositive curvature: geometric and analytic aspects. Lectures in Mathematics, ETH Zürich, Birkhäuser, Basel.
J. Jost (1997), Generalized Dirichlet forms and harmonic maps. Calc. Var. Partial Differential Equations 5, no. 1, 1-19.
W.S. Kendall (1990), Probability, convexity, and harmonic maps with small images. I. Uniqueness and fine existence. Proc. London Math. Soc. 61, 371-406.
N. Korevaar, R. Schoen (1993), Sobolev spaces and harmonic maps for metric space targets. Comm. Anal. Geom. 1, 561-569.
N. Korevaar, R. Schoen (1997), Global existence theorems for harmonic maps to non-locally compact spaces. Comm. Anal. Geom. 5, no. 2, 333-387.
M. Ledoux (2000), The geometry of Markov diffusion generators. Probability theory. Ann. Fac. Sci. Toulouse Math. (6) 9 (2000), no. 2, 305-366.
M. Ledoux, Talagrand (1991), Probability in Banach spaces. Isoperimetry and processes. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 23, Springer, Berlin.
Z.M. MA, M. Röckner (1992), Introduction to the theory of (nonsymmetric) Dirichlet forms. Universitext, Springer, Berlin.
S.T. Rachev, L. Rüschendorf (1998), Mass transportation problems. Vol. I. Theory. Probability and its Applications, Springer, New York.
M.-K. VON Renesse (2002), Comparison properties of diffusion semigroups on spaces with lower curvature bounds. PhD Thesis, Bonn.
D.W. Stroock, S.R.S. Varadhan (1979), Multidimensional diffusion processes. Grundlehren der Mathematischen Wissenschaften 233, Springer-Verlag, Berlin-New York.
K.T. Sturm (2001), Nonlinear Markov operators associated with symmetric Markov kernels and energy minimizing maps between singular spaces. Calc. Var. 12, 317-357
K.T. Sturm (2002), Nonlinear martingale theory for processes with values in metric spaces of nonpositive curvature. To appear in Ann. Prob.
K.T. Sturm (2002A), Nonlinear Markov operators, discrete heat flow, and harmonic maps between singular spaces. To appear in Potential Analysis.
K.T. Sturm (2002B), A semigroup approach to harmonic maps. Preprint, Bonn.
K. Taira (1991), Boundary value problems and Markov processes. Lecture Notes in Mathematics, 1499, Springer-Verlag, Berlin.
A. Thalmaier (1996), Martingales on Riemannian manifolds and the nonlinear heat equation. Stochastic analysis and applications, 429-440, World Sci. Publishing, River Edge, NJ.
A. Thalmaier (1996A), Brownian motion and the formation of singularities in the heat flow for harmonic maps. Probab. Theory Related Fields 105, no. 3, 335-367.
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Sturm, KT. (2003). Harmonic Map Heat Flow generated by Markovian Semigroups. In: Iannelli, M., Lumer, G. (eds) Evolution Equations: Applications to Physics, Industry, Life Sciences and Economics. Progress in Nonlinear Differential Equations and Their Applications, vol 55. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8085-5_26
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