For $y\in \mathbb R/\mathbb Z$ and $\varphi\in C([0,\infty);\mathbb R/\mathbb Z)$ let
$T_{y}(\varphi) \ := \ \inf\{t>0: \varphi_t = y \} \ \ \ $ (first time the path $\varphi$ hits $y$)
and, for each $x\in \mathbb R/\mathbb Z$, let $\mathbb P_x$ be the Wiener measure on $ C([0,\infty);\mathbb R/\mathbb Z)$ with starting point $x$.
I want to show that there exist constants $\alpha,\beta>0$ such that for every $t>0$ and $x,y\in \mathbb R/\mathbb Z$
$ \mathbb P_x (T_y \geq t) \ \leq \ \alpha e^{-\beta t} \ \ \ \ \ (*) $
I think one way to show it would be: to show first that there exist $K,K'>0$ such that
$ \inf_{x,y\in \mathbb R/\mathbb Z} \mathbb P_{x} (T_y \leq K) \geq K' $
and then iterate the Markov property to get (*).
Any suggestion?