  In this paper we prove a large deviation principle for the empirical drift of
a one-dimensional Brownian motion with self-repellence called the Edwards
model. Our results extend earlier work in which a law of large numbers,
respectively, a central limit theorem were derived. In the Edwards model a path
of length $T$ receives a penalty $e^{-\beta H_T}$, where $ H_T$ is the
self-intersection local time of the path and $\beta\in(0,\infty)$ is a
parameter called the strength of self-repellence. We identify the rate function
in the large deviation principle for the endpoint of the path as $\beta^{\frac
23} I(\beta^{-\frac 13}\cdot)$, with $I(\cdot)$ given in terms of the principal
eigenvalues of a one-parameter family of Sturm-Liouville operators. We show
that there exist numbers $0<b^{**}<b^*<\infty$ such that: (1) $I$ is linearly
decreasing on $[0,b^{**}]$; (2) $I$ is real-analytic and strictly convex on
$(b^{**},\infty)$; (3) $I$ is continuously differentiable at $b^{**}$; (4) $I$
has a unique zero at $b^*$. (The latter fact identifies $b^*$ as the asymptotic
drift of the endpoint.) The critical drift $b^{**}$ is associated with a
crossover in the optimal strategy of the path: for $b\geq b^{**}$ the path
assumes local drift $b$ during the full time $T$, while for $0\leq b<b^{**}$ it
assumes local drift $b^{**}$ during time $\frac{b^{**}+b}{2b^{**}}T$ and local
drift $-b^{**}$ during the remaining time $\frac{b^{**}-b}{2b^{**}}T$. Thus, in
the second regime the path makes an overshoot of size $\frac{b^{**}-b}{2}T$ in
order to reduce its intersection local time.
