Are known expressions for total variation distance between $N(0,\sigma^2)$ and $N(0,\sigma^2+\epsilon)$ for small $\epsilon$? The only thing I seem to find is things are expression about the mean but not if we change variance slightly.



Are there known expressions for total variation distance between $N(0,\sigma_1^2)$ and $N(0,\sigma^2)$
There's also a softer argument based on properties of the heat kernel, which applies in higher dimensions as well, in Lemma 4.9 of [this paper of Klartag](http://arxiv.org/abs/math/0605014). It shows that in $n$ dimensions, the total variation distance between centered Gaussian distributions with covariances $\alpha I\_n$ and $\beta I\_n$ is at most
$$
C \sqrt{n} \left|\frac{\beta}{\alpha} - 1\right|,
$$
where $C > 0$ is an absolute constant (which can be made explicit if you want).


