A probabilisitc solution is given by MR1469575 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 and MR2722788.

A probabilisitc solution is given by MR1469575 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 and MR2722788.

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

In the recent preprint arXiv:2304.11189, 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 $\partial_t u = \frac{1}{2} \Delta u^\alpha$.