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Nonlinear martingale theory for processes with values in metric spaces of nonpositive curvature. (English) Zbl 1017.60050

The goal of the paper is to extend the classical linear martingale theory towards discrete time processes with values in a metric space \((N,d)\) with nonpositive curvature (NPC space). This leads to a stochastic approach to harmonic maps, the nonlinear heat flow and the Dirichlet problem. The paper begins with an introduction to NPC spaces and the associated spaces \(L^r\) of random variables \(Y\): \((\Omega,\mathcal{F},P) \rightarrow (N,d)\) with \(r\)th moments, \(r \in [1,\infty]\). Given a sub-\(\sigma\)-field \({\mathcal G}\) of \(\mathcal{F}\), the nonpositive curvature of \((N,d)\) allows the definition of the conditional expectation \(E(Y|\mathcal{G})\) as the unique \(\mathcal{G}\)-measurable random variable with minimal \(L^2\)-distance to \(Y\), i.e. the barycenter of \(Y\). Using geodesics to define convex combinations of \(N\)-valued random variables, a weak and a strong law of large numbers are proved. The tower property does in general not hold for the introduced conditional expectation. In order to remedy this defect the concept of filtered conditional expectations is introduced, and the Jensen inequality for conditional and filtered conditional expectations are proved. Discrete time \(N\)-valued martingales and associated bracket processes are introduced, and several properties are proved. Among them are an optional stopping and martingale convergence theorem. Finally, nonlinear Markov properties are proved, and the connection between the associated harmonic maps and nonlinear heat flows and the martingale property are discussed. These considerations are then extended to the Dirichlet problem, which is uniquely solved using the filtered conditional expectation.

MSC:

60G42 Martingales with discrete parameter
58E20 Harmonic maps, etc.
60J05 Discrete-time Markov processes on general state spaces
60J45 Probabilistic potential theory
31C25 Dirichlet forms
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