Distribution of time spent by Brownian motion in a set

Question

Let $B_t(\omega)$ be a standard Brownian motion and let $A\in\mathcal B(\mathbb R)$ be a Borel set. I would like to find the distribution of $$Y_A(\omega):=\lambda(\{t\in[0,1]:B_t(\omega)\in A\})=\int_0^1\mathbf1_{B_t(\omega)\in A}dt.$$ I was only able to find its expected value. $$ \mathbb E[Y_A]=\int\int_0^1\mathbf1_{B_t\in A}dtd\omega=\int_0^1\int\mathbf 1_{B_t\in A}d\omega dt=\int_0^1\mathbb P[\mathcal N(0,t)\in A]dt=\int_0^1\int_A\frac1{\sqrt{2\pi t}}\exp\left(-\frac1{2t}x^2\right)dxdt\\ =\int_Ax\mathrm{erf}\left(\frac x{\sqrt{2}}\right)-|x|+\sqrt{ \frac2\pi}\exp\left(-\frac{x^2}2\right) dx $$ Is it possible to compute higher moments or the characteristic function with similar techniques? In particular I would be particularly interested in sets of the form $A=[-a,a]$ for $a>0$. In this case, $$\mathbb E[Y_{[-a,a]}]=\mathrm{erf}\left(\frac a{\sqrt{2}}\right)-a^2\mathrm{erfc}\left(\frac a{\sqrt{2}}\right)+\frac{2a}{\sqrt{ 2\pi}}\exp\left(-\frac{a^2}2\right)$$