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| Format: | Preprint |
| Published: |
2025
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| Online Access: | https://arxiv.org/abs/2503.20171 |
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| _version_ | 1866908284920791040 |
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| author | Nakashima, Makoto |
| author_facet | Nakashima, Makoto |
| contents | In [CSZ23], the authors proved the convergence of the finite dimensional time distribution of the rescaled random fields derived from the discrete stochastic heat equation of $2d$-directed polymers in random environment in the critical window. The scaling limit is called critical $2d$ stochastic heat flow (SHF).
In this paper, we will show that the critical $2d$ SHF is a continuous semimartingale. Moreover, we will consider the martingale problem associated with the critical $2d$ SHF in a similar fashion to the super Brownian motion which is one of the well-known measure valued process. Also, we define the martingale measure associated with the critical $2d$ SHF in the sense of [Wal86, Chapter 2].
The quadratic variation of the martingale measure gives information of the regularity of the critical $2d$ SHF. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_20171 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Martingale measure associated with the critical $2d$ stochastic heat flow Nakashima, Makoto Probability 60H17, 65C35, 60G44 In [CSZ23], the authors proved the convergence of the finite dimensional time distribution of the rescaled random fields derived from the discrete stochastic heat equation of $2d$-directed polymers in random environment in the critical window. The scaling limit is called critical $2d$ stochastic heat flow (SHF). In this paper, we will show that the critical $2d$ SHF is a continuous semimartingale. Moreover, we will consider the martingale problem associated with the critical $2d$ SHF in a similar fashion to the super Brownian motion which is one of the well-known measure valued process. Also, we define the martingale measure associated with the critical $2d$ SHF in the sense of [Wal86, Chapter 2]. The quadratic variation of the martingale measure gives information of the regularity of the critical $2d$ SHF. |
| title | Martingale measure associated with the critical $2d$ stochastic heat flow |
| topic | Probability 60H17, 65C35, 60G44 |
| url | https://arxiv.org/abs/2503.20171 |