Let $M$ be a connected Riemannian manifold and $x_0 \in M$. For $0 < \alpha < \frac 1 2$, let

$$H = \{ c : [0,1] \to M \mid c(0) = x_0 \text{ and } \exists C>0 \text { s.t. } d(c(s), c(t)) \le C |s-t| ^\alpha \ \forall s,t \}$$

where $d$ is the Riemannian distance on $M$. The Wiener measure $w$ is concentrated on $H$ and $w(H) \le 1$. (It is not true, in general, that $w(H) = 1$.)

If, for $C>0$, we define

$$H_C = \{ c \in H \mid d(c(s), c(t)) \le C |s-t| ^\alpha \ \forall s,t \} \ ,$$

do we know a non-trivial upper bound for $w(H_C)$ as a function of $C$?

I am trying to see how fast the measure of the complementary $H \setminus H_C$ decreases when $C$ increases.

I currently know of two such results, both in $\mathbb R^n$:

  • one is theorem 3.4.16 in Stroock's first edition of "Probability theory: an analytic view" (it is missing from the 2nd edition): it is in fact a version of Kolmogorov's continuity criterion and depends essentially on the condition that

    $$\int _H \| c(s) - c(t) \| ^r \ \mathbb d w \le A \ |t-s|^{1 + \alpha}$$

    for some $A>0$, $r \ge 1$, $\alpha > 0$ and all $s,t \in [0,1]$ - which does not hold for general Riemannian manifolds, since the heat kernel of $M$ may be too wild;

  • the other has two proofs in "Hölder norms and the support theorem for diffusions" by Ben Arous, Gradinaru and Ledoux, but the techniques therein depend heavily on the underlying linear structure of $\mathbb R^n$.


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