- Date: 18 Sep. (Tue), 20 Sep. (Thu), 25 Sep. (Tue), 27 Sep. (Thu), and 4 Oct. (Thu)
- Place: W.W. 7th-floor
- Time : 16:30-18:00
- Speaker: Vlad Bally (Université de Marne-la-Vallée)
- Title: Integration by parts formulas and regularity of probability laws
- Abstract:
One of the most important applications of Malliavin calculus is to built integration by parts formulas which themselves are used in order to study the regularity of the law of functionals on the Wiener space. The aim of my lectures is to discuss the way in which abstract integration by part formulas may be used in order to obtain such regularity results. A Örst point is to prove that a single integration by parts is already su¢ cient in order to obtain a continuous density for the law of a random variable.
This result has been proved by Malliavin itself in his book and afterwords Shigekava gave a new approach and a new proof for this result. In a paper in collaboration with L. Caramellino we gave another approach and in particular we studied Sobolev spaces associated to a probability law (a similar formalism has been used by Malliavin and by Shigekava). A second point concerns the study of functionals which are not in the domain of the Malliavin di§erential operators and so the Malliavin calculus does not apply. We prove that however one may obtain the regularity results by approximating the functional at hand by functionals which are “regular” and then by establishing a sweated equilibrium between the speed of approximation on one hand and the blow up of the weights which appear in the integration by parts formulas for the approximating functionals on the other hand. It turns out that this procedure is closely linked with the technics employed in the interpolation space theory.
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