Another useful way to study local time is related to the very useful occupation formula (the meaning of it is obvious if you think a little bit): $\int_0^t g(W_s) ds = \int_{\mathbb R} g(x) L(x,t) dx $.

Putting $g(x) = e^{izx}$: $$ \int_{\mathbb R} e^{izx} L(x,t) dx = \int_0^t e^{iz W_s}ds. $$ Now the lhs is the Fourier transform of $L$; inverting it, we get $$ L(x,t) = \frac{1}{2\pi} \int_{\mathbb R} e^{-izx} \int_0^t e^{iz W_s} ds\\, dy = \frac{1}{2\pi} \int_{\mathbb R} \int_0^t e^{iz (W_s-x)} ds\\, dy. $$ From here one can e.g. more or less easily find the moments of $L$.

Another useful way to study local time is related to the very useful occupation formula (its meaning is obvious if you think a little bit): $\int_0^t g(W_s) ds = \int_{\mathbb R} g(x) L(x,t) dx $.

Putting $g(x) = e^{izx}$: $$ \int_{\mathbb R} e^{izx} L(x,t) dx = \int_0^t e^{iz W_s}ds. $$ Now the lhs is the Fourier transform of $L$; inverting it, we get $$ L(x,t) = \frac{1}{2\pi} \int_{\mathbb R} e^{-izx} \int_0^t e^{iz W_s} ds\\, dz = \frac{1}{2\pi} \int_{\mathbb R} \int_0^t e^{iz (W_s-x)} ds\\, dz. $$ From here one can e.g. more or less easily find the moments of $L$.