Let $\mu\_0$ be standard Wiener measure on $C[0,T]$. Let $F\_C(t)=Ct$ and consider the corresponding Girsanov measure
$$\frac{d\mu\_C}{d\mu\_0}=\exp\left(CB(T)-\frac12C^2T\right)$$


What is the weak\* limit of this measure as $C\to \infty$? We know that the space of signed Borel measures on $C[0,T]$ is compact so we have that $\mu\_C$ has a convergence subsequence. I suspect it converges to the zero measure. Because for any $B(T)$ and any $T$, the limit of the density is $0$.



What is the weak* limit of the Girsanov density associated to drift $F(t)=Ct$?
The probability measure $\mu\_C$ does not converge to any measure as $C\to\infty$. One way to see this is to note that
$$f\_C(u):=\int\_{C[0,T]} e^{iu x(T)}\,\mu\_C(dx)=e^{iCTu-Tu^2/2}$$
for all real $u$, because $f\_C$ is the characteristic function of the random variable $B(T)+CT$. So, for any sequence of real numbers $C\_n$ converging to $\infty$, the sequence of the functions $f\_{C\_n}$ does not converge to any function and thus the sequence of the measures $\mu\_{C\_n}$ does not converge to any measure.


Another way to come to the same conclusion is to note that for the cdf $F\_C$ of $B(T)+CT$ and all real $z$ we have $F\_C(z)\to0$ as $C\to\infty$; that is, $F\_C$ converges to the cdf of the zero measure on $\mathbb R$. However, a sequence $(\nu\_k)$of probability measures can only converge to a probability measure $\nu$, because then $1=\int1\,d\nu\_k\to\int1\,d\nu$.


