arXiv:2606.15203v1

The existence of invariant sublinear expectations for $G$ -SDEs

Danyang Zhao, Huaizhong Zhao

math.PR60G6560G1037A50

Paper metadata

Original submission: June 13, 2026
Claimed MSC: 60G65, 60G10, 37A50
arXiv categories: math.PR
Paper license: arXiv non-exclusive distribution license ↗
Comments: 35 pages, this work has been accepted by the SIAM Journal on Mathematical Analysis

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 $G\text{-}$ diffusions by studying the Lyapunov function and the local Doeblin condition. One is the $G\text{-}$ Brownian motion on the unit circle which is the case studied in Feng and Zhao, but with the new method. Another is the multidimensional $G\text{-}$ SDEs on the whole space $\mathbb{R}^d$ . We establish, for the first time in the literature, the existence of the invariant sublinear expectation for $G\text{-}$ SDEs under the non-degenerate and weakly dissipative assumption. For this, we prove that for a class of $G\text{-}$ SDEs, the $G\text{-}$ expectation can be represented as the supremum of the semigroup of a family of SDEs, of which the regularity is obtained by considering the Bismut-Elworthy-Li formula and the Denis-Hu-Peng representation for the distribution of $G\text{-}$ Brownian motions.

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Audited against arXiv v1

Not a correctness certificate. A “Correct” result may include yellow typos or minor formal corrections that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

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Detailed mathematical audit

Generated August 18, 2026

01Statements3 reported findingsContains unsupported statements

The abstract Harris-type fixed-point theorem is plausibly repairable, but the principal G-SDE invariant-expectation theorem is not able to be verified. Its proof depends on Lemma 4.6, which incorrectly replaces an adapted random diffusion coefficient by a deterministic coefficient with the same expected covariance. A simple uniformly elliptic adapted control gives a non-Gaussian law and disproves that lemma.

Theorem 4.13Not able to verify

Existence and uniqueness for the weakly dissipative G-SDE are not established

Pages 26–28 · Theorem 4.13 · arXiv:2606.15203v1

The Lyapunov estimate verifies the drift part of the abstract theorem. The local Doeblin bound, however, is obtained by applying density estimates after Lemma 4.6 has replaced every adapted volatility control by a deterministic one. Lemma 4.6 is false, as shown in the Proofs findings, so that density reduction and the uniform minorization do not follow. Uniformly elliptic controlled diffusions may admit other minorization arguments, but none is proved here. The audit therefore does not claim Theorem 4.13 is false; it records that the supplied proof does not verify it.

Theorems 3.8 and 3.12Correct

The abstract invariant-sublinear-expectation conclusion has a repairable convergence proof

Pages 6–13 · discrete and continuous Harris theorems · arXiv:2606.15203v1

The drift and minorization assumptions yield contraction of the weighted Lipschitz seminorm and therefore contraction of pushed-forward sublinear expectations modulo constants. This gives a Cauchy sequence and uniqueness by the same contraction. The printed independence-of-skeleton argument contains a false norm estimate, but the conclusion is recovered by comparing common multiples of two skeletons and then using the consecutive skeletons $n$ and $n+1$ to obtain one-step invariance. The continuous-time result follows by the bounded short-time drift estimate exactly as in the paper.

G-Brownian motion on the circleCorrect

The compact nondegenerate example satisfies the abstract assumptions

Pages 14–17 · Section 4.1 and Theorem 4.2 · arXiv:2606.15203v1

On the compact circle the Lyapunov condition is immediate, and nondegeneracy gives a uniform positive lower density over a fixed time. These facts supply the minorization without using the false deterministic-volatility reduction needed later on $\mathbb R^d$ .

02Proofs3 reported findingsContains incorrect or incomplete proofs

Lemma 4.6 is false for adapted volatility controls and invalidates the main G-SDE minorization proof. The abstract discrete convergence proof also contains a false weighted-norm contraction, although that earlier theorem has a verified repair.

Lemma 4.6Incorrect as written

An adapted diffusion coefficient cannot be replaced by its deterministic mean covariance

Pages 19–20 · Lemma 4.6 and equations (4.17)–(4.19) · arXiv:2606.15203v1

For an arbitrary adapted $\theta$ , the process need not define the Markov semigroup asserted in the proof, and the generator cannot replace $\theta_t\theta_t^T$ by its expectation independently of the state. A concrete one-dimensional counterexample has $b=0$ , $\Theta=\{1,2\}$ , $\theta_t=1$ for $t\leq1/2$ , and, for $t>1/2$ , $\theta_t=1$ when $W_{1/2}\geq0$ and $\theta_t=2$ otherwise. Then $X_1=W_{1/2}+\theta_{(1/2,1]}(W_1-W_{1/2})$ has a nonzero third moment, because $3\operatorname{Var}(W_1-W_{1/2})\,\mathbb E[W_{1/2}\theta_{(1/2,1]}^2]\neq0$ . Every stochastic integral with deterministic coefficient and zero drift is centered Gaussian and has zero third moment. Thus no deterministic $\bar\theta$ can give the claimed representation, even though the example is uniformly elliptic and satisfies (H1).

Proof of Theorem 4.13Incomplete as written

The uniform Doeblin lower bound uses the false representation lemma

Pages 27–28 · verification of Assumption B-prime · arXiv:2606.15203v1

The proof applies an Aronson-type density lower bound to the deterministic diffusion furnished by Lemma 4.6 and then takes an infimum over adapted controls. Since that deterministic representation fails, the displayed density is not the law of the controlled process and its constants do not establish the required infimum. A repair would need a density or coupling theorem uniform over all progressively measurable uniformly elliptic controls, stated and proved at the exact bounded-measurable test-function level used here.

Proof of Theorem 3.6Incorrect as written · verified repair

Weighted supremum norms do not contract by the Lipschitz contraction factor

Pages 10–11 · independence of the discrete skeleton · arXiv:2606.15203v1

The proof bounds $\|\overline T^{mk}\varphi\|_\beta$ by $\bar\alpha^m\|\varphi\|_\beta$ . This is false even for $\varphi\equiv1$ , because a Markov operator preserves constants. What contracts is the Lipschitz seminorm, or equivalently the distance between two expectation functionals after constants cancel. Using that metric, limits along two skeletons agree on their common-multiple subsequence; the consecutive skeletons $n$ and $n+1$ then imply one-step invariance, and the drift bound controls the remaining residues. This supplies a verified replacement proof of the abstract theorem.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2606.15203v1
Authors listed: Danyang Zhao, Huaizhong Zhao
Audit date: August 18, 2026

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The existence of invariant sublinear expectations for GGG-SDEs

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Existence and uniqueness for the weakly dissipative G-SDE are not established

The abstract invariant-sublinear-expectation conclusion has a repairable convergence proof

The compact nondegenerate example satisfies the abstract assumptions

An adapted diffusion coefficient cannot be replaced by its deterministic mean covariance

The uniform Doeblin lower bound uses the false representation lemma

Weighted supremum norms do not contract by the Lipschitz contraction factor

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The existence of invariant sublinear expectations for $G$ -SDEs