# Stochastic partial differential equation

$\partial_t u = \Delta u + \xi\;,$
where $\xi$ denotes space-time white noise and $\Delta$ is the Laplacian. In one space dimension, solutions to this equation are only almost 1/2-Hölder continuous in space and 1/4-Hölder continuous in time. For dimensions two and higher, solutions are not even function-valued, but can be made sense of as random distributions.