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Logarithmic Sobolev inequalities on path spaces. (Inégalités de Sobolev logarithmiques sur un espace de chemins.) (French) Zbl 0826.46023

Summary: Suppose that \(W_o(M)\) is the space of paths \(\gamma: [0, 1]\to M\) on a complete connected Riemannian manifold \(M\) of dimension \(n\) such that \(\gamma(0)= o\in M\). We prove a logarithmic Sobolev inequality on \(W_o(M)\): If \(\text{Ric}_M\geq - c\) for a nonnegative constant \(c\), then \[ \int_{W_o(M)} F^2\log |F|d\nu\leq C_M|DF|^2+ |F|^2\log |F| \] with \(C_M\leq (1+ ce^c)^2\).

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
58D15 Manifolds of mappings
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