The aim of these notes is to provide an introduction to the Malliavin calculus and its application to the regularity of the solutions of a class of stochastic partial differential equations. The Malliavin calculus is a differential calculus on a Gaussian space which has been developed from the probabilistic proof by Malliavin of H¨ormander's hypoellipticity theorem (see [8]). In the next section we present an introduction to the Malliavin calculus, and we derive the main properties of the derivative and divergence operators. Section 3 is devoted to establish the main criteria for the existence and regularity of density for a random variable in a Gaussian space.
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Nualart, D. (2009). Application of Malliavin Calculus to Stochastic Partial Differential Equations. In: Khoshnevisan, D., Rassoul-Agha, F. (eds) A Minicourse on Stochastic Partial Differential Equations. Lecture Notes in Mathematics, vol 1962. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-85994-9_3
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