A CLT for the L^{2} modulus of continuity of Brownian local time

Let $\{L_t^x;(x,t)\in R^1\times R_{+}^1\}$ denote the local time of Brownian motion and $${\alpha }_t:={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}(L_t^x{)}^{2}dx\mathrm{.}$$ Let $\eta =N(0,1)$ be independent of ${\alpha }_t$. For each fixed $t$ $$\frac{{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}(L_t^{x+h}-L_t^x{)}^{2}dx-4ht}{h^{3\mathrm{/}2}}(\frac{64}{3}{)}^{1\mathrm{/}2}\sqrt{{\alpha }_t}\eta ,$$ as $h\text{rar}0$. Equivalently $$\frac{{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}(L_t^{x+1}-L_t^x{)}^{2}dx-4t}{t^{3\mathrm{/}4}}(\frac{64}{3}{)}^{1\mathrm{/}2}\sqrt{{\alpha }_1}\eta ,$$ as $t\text{rar}\mathrm{\infty }$.