Let $f(t)$, $t \geq 0$ be a smooth function with $f(0) = 0$ and let $B(t)$, $t \geq 0$ be a brownian motion. Let $P$ and $Q$ be two measures on $C[0,1]$ corresponding to respectively, $B(t)$, $t \geq 0$ and $X(t) = f(t) + B(t)$, $t \geq 0$. i.e. for a Borel set $A \subset C[0,1]$, $P(A)$ is the probability that the path $B(.)$ is in $A$ and $Q(A)$ is the probability that the path $X(.)$ is in $A$.
a) is $Q$ absolutely continuous with respect to $P$?
b) Define the local time $L_t(x)$ of $X(t)$ at $x$ by
$\int_A L_t(x)dx = \int_0^t1_A(X(s))ds$ with $A$ open.
is $L_t(x)$ an a.s continuous function of $t$?
Now I think the answer to part a is Yes by using Girsanvo change of measures we can just say that $Q$ equals the integral of some smooth function with respect to $dP$.
I don't know how to do partb though. I appreciate someone explaining to me.