A probabilisitc solution is given by [MR1469575][1] a nonlinear diffusion
$$
  Y_t = Y_0 + \int_0^t u^{\frac{m-1}{2}}(s,Y_s)\; \mathrm{d}W_s , \qquad \mathrm{law}(Y_0) = u(0,\cdot) 
$$
then
$$ \mathrm{law}(Y_t) = u(t,\cdot) . $$
This is true for a general class of nonlinear diffusion equations. The best references I've found are [MR1775228][2] and [MR2722788][3].

**update 22.05.2023:** ***A rescaled zero-range process for the porous medium equation*** 

In the recent preprint [arXiv:2304.11189][4], Gess and Heydecker show that the nearest neighbor zero-range process with jump rates $k^\alpha$ on the torus of size $N$ converges after some suitable rescaling of particle size and time to the solution of the PME $\partial_t u = \frac{1}{2} \Delta u^\alpha$. 


  [1]: http://www.ams.org/mathscinet-getitem?mr=1469575
  [2]: http://www.ams.org/mathscinet-getitem?mr=1775228
  [3]: http://www.ams.org/mathscinet-getitem?mr=2722788
  [4]: https://arxiv.org/abs/2303.11289