A Geometric Extension of the It\^o-Wentzell and Kunita's Formulas

Aythami Bethencourt de León; So Takao

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arXiv · 2311.04439

A Geometric Extension of the It\^o-Wentzell and Kunita's Formulas

Aythami Bethencourt de León·So Takao

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Abstract

We extend the It\^o-Wentzell formula for the evolution along a continuous semimartingale of a time-dependent stochastic field driven by a continuous semimartingale to tensor field-valued stochastic processes on manifolds. More concretely, we investigate how the pull-back (respectively, the push-forward) by a stochastic flow of diffeomorphisms of a time-dependent stochastic tensor field driven by a continuous semimartingale evolves with time, deriving it under suitable regularity conditions. We call this result the Kunita-It\^o-Wentzell (KIW) formula for the advection of tensor-valued stochastic processes. Equations of this nature bear significance in stochastic fluid dynamics and well-posedness by noise problems, facilitating the development of certain geometric extensions within existing theories.

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BibTeX RIS

Aythami Bethencourt de León, So Takao. 2023-11-08. A Geometric Extension of the It\^o-Wentzell and Kunita's Formulas. https://arxiv.org/abs/2311.04439

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