  We consider Funaki's model of a random string taking values in R^d. It is
specified by the following stochastic PDE, du = u_{xx} + W, where W=W(x,t) is
two-parameter white noise, also taking values in R^d. We study hitting
properties, double points, and recurrence. The main difficulty is that the
process has the Markov property in time, but not in space. We find: (1) The
string hits points if d<6. (2) For fixed t, there are points x,y such that
u(t,x)=u(t,y) iff d < 4. (3) There exist points t,x,y such that u(t,x)=u(t,y)
iff d < 8. (4) There exist points s,t,x,y such that u(t,x)=u(s,y) iff d < 12.
(5) The string is recurrent iff d < 7.
