In this talk we give a survey about some common methods to show the absolute continuity of the probability law of the solution to SDEs and SPDEs. The most-used approach to this problem is to show that the random variables that form the solution are Malliavin differentiable and then apply an absolute continuity criterion such as the Bouleau-Hirsch criterion, see [4] for an overview. However, in recent years other approaches have appeared that also work under less restrictive hypotheses. We will present the following three approaches to show that a random variable X is absolutely continuous:

1. the approach in [3] that uses the auxiliary random variable G := <DX, -DL^{-1}X>_H , where D is the Malliavin derivative operator and L is Ornstein-Uhlenbeck operator,

2. the approach in [2] that uses the characteristic function of X, and

3. the approach in [1], where one uses Besov-space methods to show the existence of a density.

To each of these approaches we will also provide some specific examples where it works best.

References

[1] A. Debussche and N. Fournier. Existence of densities for stable-like driven SDE’s with Hölder continuous coefficients. J. Funct. Anal., 264(8):1757–1778, 2013.

[2] N. Fournier and J. Printems. Absolute continuity of some one-dimensional processes. Bernoulli, 16:343360, 2010.

[3] I. Nourdin and F. Viens. Density formula and concentration inequalities with Malliavin calculus. Electron. J. Probab., 14(78):2287–2309, 2009.

[4] D. Nualart. The Malliavin Calculus and Related Topics. Springer Verlag, Berlin, 2nd edition, 2006.