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

(UCSD Thesis, August 1996) )

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

Abstract

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 $W_o(M).$

 08/15/2016 02:53 PM