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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2409.00715 |
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| _version_ | 1866917887246073856 |
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| author | Watanabe, Takayoshi |
| author_facet | Watanabe, Takayoshi |
| contents | We deal with Malliavin calculus on the $L^2$ space of the $W^*$-algebra generated by fermion fields (the Clifford algebra). First, we verify the product formula for multiple integrals in Itô-Clifford calculus, which is Itô calculus on the Clifford algebra. Using this product formula, we can define the derivation operator and the divergence operator. Anti-symmetric Malliavin calculus thus constructed has properties similar to those of usual Malliavin calculus. The derivation operator and the divergence operator satisfy the canonical anti-commutation relations, and the divergence operator serves as an extension of the Itô-Clifford stochastic integral, satisfying the Clark-Ocone formula. Subsequently, using this calculus, we consider the concentration inequality, the logarithmic Sobolev inequality, and the fourth-moment theorem. As for the logarithmic Sobolev inequality, only a weaker result is obtained. On the other hand, as for the concentration inequality, we obtain results that are almost similar to those of the usual case; moreover, the results concerning the fourth-moment theorem imply that, unlike in the case of the usual Brownian motion, convergence in distribution cannot be deduced from the convergence of the fourth moment. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_00715 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Malliavin calculus on the Clifford algebra Watanabe, Takayoshi Probability We deal with Malliavin calculus on the $L^2$ space of the $W^*$-algebra generated by fermion fields (the Clifford algebra). First, we verify the product formula for multiple integrals in Itô-Clifford calculus, which is Itô calculus on the Clifford algebra. Using this product formula, we can define the derivation operator and the divergence operator. Anti-symmetric Malliavin calculus thus constructed has properties similar to those of usual Malliavin calculus. The derivation operator and the divergence operator satisfy the canonical anti-commutation relations, and the divergence operator serves as an extension of the Itô-Clifford stochastic integral, satisfying the Clark-Ocone formula. Subsequently, using this calculus, we consider the concentration inequality, the logarithmic Sobolev inequality, and the fourth-moment theorem. As for the logarithmic Sobolev inequality, only a weaker result is obtained. On the other hand, as for the concentration inequality, we obtain results that are almost similar to those of the usual case; moreover, the results concerning the fourth-moment theorem imply that, unlike in the case of the usual Brownian motion, convergence in distribution cannot be deduced from the convergence of the fourth moment. |
| title | Malliavin calculus on the Clifford algebra |
| topic | Probability |
| url | https://arxiv.org/abs/2409.00715 |