We will introduce parabolic differential equations driven by white noise in time. We will be mostly interested in nonlinear parabolic equations with a nonlinearity π in the leading order term and a noise ξ that is white not only in time but also in space. The latter limits the space dimension to one, leading to $$\partial_tu-\partial_x^2\pi(u)=\xi$$.

We are interested in the path-wise regularity of solutions to such equations. In case of our model problem, a scaling argument suggests that the solutions are Hölder continuous (with almost exponent ½ in space and almost exponent ¼ in time). This is also the regularity in the linear case.
We shall show that this is indeed true. The argument relies on the following ingredients:

• On the stochastic side:
  1. Arguments typical for stochastic differential equation (Martingale arguments) that give second-moment regularity estimates.
  2. Concentration of measure arguments on the level of the space-time white noise (Malliavin derivative) that upgrade the low-moment regularity results to Gaussian moments.
• On the deterministic side:
  1. The Ḣ^-1-contraction principle for nonlinear parabolic equations of the form $$\partial_tu-\partial_x^2\pi(u)=0$$
  2. Campanato-type arguments for a Schauder theory for non-constant coefficient parabolic equations of the form $$\partial_tu-\partial_x^2(au)=f$$.

Hence despite the specifics of the model problem, the arguments are fairly general and it is thus a good excuse for introducing the above-mentioned concepts.