We study the porous medium equation (PME) in one space dimension in the presence of additive nonconservative white noise and interpreted as a stochastic growth equation for the height field of an interface. We predict the values of the two growth exponents $\alpha$ and $\beta$ using the functional renormalization group. Extensive numerical simulations show agreement with the predicted values for these exponents; however, they also show anomalous scaling with an additional local exponent ${\alpha }_{\mathrm{loc}}$ as well as multiscaling originating from broad distributions of local height differences. The stationary measure of the stochastic PME is found to be well described by a random walk model related to a Bessel process. This model allows for several predictions about the multiscaling properties.

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• Continuous-time random walk
• Critical exponents
• Fractional dimension
• Growth processes
• Interface & surface thermodynamics
• Interfaces
• Surface growth
• Finite-size scaling
• Fokker–Planck equation
• Langevin algorithm
• Markovian processes
• Renormalization group
• Stochastic analysis
• Stochastic differential equations