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The existence of invariant sublinear expectations for G-SDEs

  • Danyang Zhao
  • Huaizhong Zhao
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Abstract

In this paper, we study the existence of invariant sublinear expectations of Markovian semigroups on sublinear expectation spaces. To achieve this, we establish a complete metric space of sublinear expectations, on which we extend Harris' method to the nonlinear setting on the convergence of sublinear semigroups. We then explore two cases of G−diffusions by studying the Lyapunov function and the local Doeblin condition. One is the G−Brownian motion on the unit circle which is the case studied in Feng and Zhao(2021), but with the new method. Another is the multidimensional G−SDEs on the whole space ℝd. We establish, for the first time in the literature, the existence of the invariant sublinear expectation for G−SDEs under the non-degenerate and weakly dissipative assumption. For this, we prove that for a class of G−SDEs, the G−expectation can be represented as the supremum of the semigroup of a family of SDEs, of which the regularity is obtained by considering { oscillations and coupling of the nonlinear Markovian semigroup,} and the Denis-Hu-Peng representation for the distribution of G−Brownian motions.

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Cite

BibTeXRIS
BibTeX
@misc{zhao2026existence,
  title = {{The existence of invariant sublinear expectations for $G$-SDEs}},
  author = {Danyang Zhao and Huaizhong Zhao},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2606.15203},
  eprint = {2606.15203},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.15203},
}
RIS
TY  - UNPB
TI  - The existence of invariant sublinear expectations for G-SDEs
AU  - Danyang Zhao
AU  - Huaizhong Zhao
PY  - 2026
DA  - 2026/11/02
PB  - arXiv
DO  - 10.48550/arxiv.2606.15203
UR  - https://arxiv.org/abs/2606.15203
AB  - In this paper, we study the existence of invariant sublinear expectations of Markovian semigroups on sublinear expectation spaces. To achieve this, we establish a complete metric space of sublinear expectations, on which we extend Harris' method to the nonlinear setting on the convergence of sublinear semigroups. We then explore two cases of G−diffusions by studying the Lyapunov function and the local Doeblin condition. One is the G−Brownian motion on the unit circle which is the case studied in Feng and Zhao(2021), but with the new method. Another is the multidimensional G−SDEs on the whole space ℝ^d. We establish, for the first time in the literature, the existence of the invariant sublinear expectation for G−SDEs under the non-degenerate and weakly dissipative assumption. For this, we prove that for a class of G−SDEs, the G−expectation can be represented as the supremum of the semigroup of a family of SDEs, of which the regularity is obtained by considering { oscillations and coupling of the nonlinear Markovian semigroup,} and the Denis-Hu-Peng representation for the distribution of G−Brownian motions.
ER  -