Carolyn Cross
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"Differentials of Measure-Preserving Flows on Path Space" by Carolyn Cross.

(UCSD Thesis, August 1996) )

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The manuscript is available as a PDF file (546K)

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Let $W=\{\omega :[0,1]\to {\bf R}^n|\omega\mbox{\rm \ is continuous} \},$ equipped with Wiener measure. The classical Cameron-Martin theorem states that the mapping $(\omega\to\omega +h)$ of $W$ to itself (for $h\in W$) preserves the measure up to a density if and only if $h\in H=\{h\in W|h(0)=0,$ $\int_0^1|h'(s)|^2ds<\infty \}$.

Bruce Driver has proved an analogous result for the space $W_o(M)$ of continuous paths on a compact manifold $M$ with a fixed base point $o\in M.$ Let $C^1\equiv \{h\in C^1([0,1],T_oM)| h(0)=0\},$ the space of once-continuously differentiable paths in $T_oM$, starting at the origin. Driver constructed a ``natural'' vector field $X^h$ corresponding to each $h\in C^1$, and showed that the induced flow $t\to\sigma^h(t ,\omega )$ starting at a ``generic'' path $\omega\in W_o(M)$ exists, and that the map $\sigma^h(t,\cdot ):W_o(M)\to W_o(M)$ preserves Wiener measure up to a density.

In my thesis I first generalize Driver's construction of measure-preserving flows to a slightly larger class $V$ of vector fields on $W.$ These are functions $Y:W\to W$ of the form \[Y(\omega )(s)=\int_0^sC(\omega )(\bar {s})d\omega (\bar {s})+\int_ 0^sR(\omega )(\bar {s})d\bar {s}\] where, roughly speaking, $C$ takes values in the skew-symmetric matrices and $R(\omega )$ is bounded by a ~``nice'' function of $\omega .$

I then show that members of $V$ generate flows which are ``smooth'' in their starting path, i.e., differentiable via any vector field in $V.$

The proof uses a modified Picard iterates method to solve a differential equation including a term with an unbounded linear operator.

The second half of my thesis is devoted to the ``geometric'' result that Driver's flows are differentiable in their starting paths. This result is proved for both the ``transferred'' flow in $W$ and the original flow in $W_o(M).$

The $W$ case is proved by showing that the class $V$ above contains Driver's ``transferred'' vector fields on $W,$ i.e., $Y^ h\in V$ for all $h\in C^1.$ Thus the result in Part I implies that the ``transferred'' flow $w^h(t,\omega )$ generated by $Y^h$ on $W$ is ``differentiable'' in its starting path $\omega$ via any of the vector fields $Y^k,$ for $k\in C^1.$ I then use certain smoothness properties of the stochastic development map to transfer this result to $W_o(M).$

11/26/2018 10:37 AM