arXiv 0903.38912009-03-23math.PRmath-phmath.MP

Wiener空间上自适应移位的熵、可逆性和变分计算

Entropy, Invertibility and Variational Calculus of the Adapted Shifts on Wiener space

Ali Süleyman Üstünel

摘要：

在这项工作中，我们研究的充要条件的正随机变量，其期望下的Wiener测度是一个，被表示为Radon-Nikodym导数的Wiener测度的形象下的适应扰动的身份与相关的新息过程的帮助下。证明了新息猜想成立的充要条件是原过程几乎必然可逆。给出了总质量为1的正随机变量的单位元扰动可逆性和可表示性的变分刻画。我们证明了满足Girsanov定理的恒等式$U=I_W+u$的自适应扰动是可逆的当且仅当$u$的动能等于$U$作用于Wiener测度$μ$所诱导的测度的熵，换句话说，$U$是可逆的当且仅当$$ \half \int_W|u|_H^2dμ=\int_W \frac{dUμ}{dμ}\log\frac{dUμ}{dμ}dμ>. $$否则l.h.s.总是严格大于相对湿度研究了它与Monge-Kantorovitch测度输运的关系。最后给出了这些结果在一个与大偏差有关的变分问题中的应用。

英文摘要：

In this work we study the necessary and sufficient conditions for a positive random variable whose expectation under the Wiener measure is one, to be represented as the Radon-Nikodym derivative of the image of the Wiener measure under an adapted perturbation of identity with the help of the associated innovation process. We prove that the innovation conjecture holds if and only if the original process is almost surely invertible. We also give variational characterizations of the invertibility of the perturbations of identity and the representability of a positive random variable whose total mass is equal to unity. We prove in particular that an adapted perturbation of identity $U=I_W+u$ satisfying the Girsanov theorem, is invertible if and only if the kinetic energy of $u$ is equal to the entropy of the measure induced with the action of $U$ on the Wiener measure $μ$, in other words $U$ is invertible iff $$ \half \int_W|u|_H^2dμ=\int_W \frac{dUμ}{dμ}\log\frac{dUμ}{dμ}dμ>. $$ otherwise the l.h.s. is always strictly greater than the r.h.s. The relations with the Monge-Kantorovitch measure transportation are also studied. An application of these results to a variational problem related to large deviations is also given.

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