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Let $g_{\mathbb{R}^n}$ denote the usual Euclidean metric on $\mathbb{R}^n$ and let $B_g(t)$ denote the Brownian motion associated to a complete metric $g$ on $\mathbb{R}^n$. Consider a Brownian motion in $\mathbb{R}^n$ starting at the origin. Let us consider $B_1$, the ball of radius $1$ (with respect to $g_{\mathbb{R}^n}$), and an open set $V \subset B_{1/2}$. Now, consider a complete metric $\tilde{g}$ on $\mathbb{R}^n$ such that $\tilde{g} = g_{\mathbb{R}^n}$ outside $B_1$, and $(1 - \varepsilon)g \leq \tilde{g} \leq (1 + \varepsilon)g$ inside $B_1$, where $\varepsilon$ is a small fixed number.

Let $\mathbb{P}_g(t)$ denote the probability that a particle undergoing Brownian motion with respect to the metric $g$ is within $V$ by time $t$. It seems clear from intuition that there exist constants $C_1(\varepsilon), C_2(\varepsilon)$ such that $$C_1(\varepsilon) \mathbb{P}_{g_{\mathbb{R}^n}}(t) \leq \mathbb{P}_{\tilde{g}}(t)\leq C_2(\varepsilon)\mathbb{P}_{g_{\mathbb{R}^n}}(t).$$

How can I prove this rigorously?

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