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 $\mathbb R^n$*.