  Motivated by Girsanov's nonuniqueness examples for SDEs, we prove
nonuniqueness for the parabolic stochastic partial differential equation (SPDE)
\[\frac{\partial u}{\partial t}=\frac{\Delta}{2}u(t,x)
+\bigl|u(t,x)\bigr|^{\gamma}\dot{W}(t,x),\qquad u(0,x)=0.\] Here $\dot{W}$ is a
space-time white noise on ${\mathbb {R}}_+\times {\mathbb {R}}$. More
precisely, we show the above stochastic PDE has a nonzero solution for
$0<\gamma<3/4$. Since $u(t,x)=0$ solves the equation, it follows that solutions
are neither unique in law nor pathwise unique. An analogue of Yamada-Watanabe's
famous theorem for SDEs was recently shown in Mytnik and Perkins [Probab.
Theory Related Fields 149 (2011) 1-96] for SPDE's by establishing pathwise
uniqueness of solutions to \[\frac{\partial u}{\partial
t}=\frac{\Delta}{2}u(t,x)+\sigma \bigl(u(t,x)\bigr)\dot{W}(t,x)\] if $\sigma$
is H\"{o}lder continuous of index $\gamma>3/4$. Hence our examples show this
result is essentially sharp. The situation for the above class of parabolic
SPDE's is therefore similar to their finite dimensional counterparts, but with
the index $3/4$ in place of $1/2$. The case $\gamma=1/2$ of the first equation
above is particularly interesting as it arises as the scaling limit of the
signed mass for a system of annihilating critical branching random walks.
