Topics in stochastic partial differential equations
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
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:
- Arguments typical for stochastic differential equation (Martingale arguments) that give second-moment regularity estimates.
- 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:
- The Ḣ-1-contraction principle for nonlinear parabolic equations of the form $$\partial_tu-\partial_x^2\pi(u)=0$$
- Campanato-type arguments for a Schauder theory for non-constant coefficient parabolic equations of the form $$\partial_tu-\partial_x^2(au)=f$$.