A CLT for the L2 modulus of continuity of Brownian local time

Abstract

Let $\{L_t^x; (x, t)∈R^1×R_+^1\}$ denote the local time of Brownian motion, and $$α_t:=∫_{−∞}^∞(L_t^x)^2 dx.$$ Let $η=N(0, 1)$ be independent of $α_t$. For each fixed $t$, $$\frac{\int_{-\infty}^{\infty}(L_{t}^{x+h}-L_{t}^{x})^{2}\,dx-4ht}{h^{3/2}}\stackrel{\mathcal{L}}{\rightarrow}\biggl(\frac{64}{3}\biggr)^{1/2}\sqrt{\alpha_{t}}\eta$$ as $h→0$. Equivalently, $$\frac{\int_{-\infty}^{\infty}(L^{x+1}_{t}-L^{x}_{t})^{2}\,dx-4t}{t^{3/4}}\stackrel{\mathcal{L}}{\rightarrow}\biggl(\frac{64}{3}\biggr)^{1/2}\sqrt{\alpha_{1}}\eta$$ as $t→∞$.