Simulating nondeterministic space-bounded computation using randomness
My comment was incorrect, and in fact the answer to your question is YES. From Wikipedia: 


  Suppose C is the complexity class of
  problems solvable in logarithmithic
  space with probabilistic Turing
  machines that never accept incorrectly
  but are allowed to reject incorrectly
  less than 1/3 of the time; this is
  called one-sided error. The constant
  1/3 is arbitrary; any x with 0 ≤ x &lt;
  1/2 would suffice.
  
  It turns out that C = NL. Notice that
  C, unlike its deterministic
  counterpart L, is not limited to
  polynomial time, because although it
  has a polynomial number of
  configurations it can use randomness
  to escape an infinite loop. If we do
  limit it to polynomial time, we get
  the class RL, which is contained in
  but not known or believed to equal NL.