AI 中文总结
研究 \(\mathbb{R}^d\) 上由超线性噪声驱动的 McKean-Vlasov 随机分数阶 \((α,p)\)-拉普拉斯方程，用单调方法和区域扩张论证建立全局适定性，利用广义弱收敛方法等建立一致大偏差原理，克服相关困难并控制扩散项。
AI 中文摘要
研究了一类广泛的 McKean-Vlasov 随机非局部分数阶 \((α,p)\)-拉普拉斯方程在全空间 \(\mathbb{R}^d\) 上的全局适定性和一致大偏差原理。该方程由超线性乘性噪声驱动，\(α \in (0,1)\) 且 \(p>2\)。通过单调方法和区域扩张论证建立全局适定性。在扩散项增长的附加条件下，利用 Salins 发展的广义弱收敛方法建立 Freidlin-Wentzell 和 Dembo-Zeitouni 一致大偏差原理。结合一致尾端估计思想、伪单调技术和 Arzelà-Ascoli 定理证明受控方程解算子的弱到强连续性，克服因 Sobolev 嵌入非紧性和分数阶 \((α,p)\)-拉普拉斯算子非线性带来的困难，并用耗散漂移项和代数不等式仔细控制超线性增长的扩散项。
英文摘要
The global-in-time well-posedness and uniform large deviation principles (LDPs) are investigated for a wide class of Mckean-Vlasov stochastic non-local fractional $(α,p)$-Laplacian equations with $α\in (0,1)$ and $p>2$ driven by superlinear multiplicative noise defined on the whole space $\mathbb{R}^d$, where the non-local nonlinear fractional $(α,p)$-Laplace operator is defined by a singular, symmetrical and translation invariant kernel function, the distribution-dependent drift terms have arbitrary polynomial growth and the distribution-dependent diffusion terms have superlinear growth. The global-in-time well-posedness is established under these conditions by using the monotone method and a domain expansion argument. Under additional conditions on the growth of diffusion terms, we establish the Freidlin-Wentzell and Dembo-Zeitouni uniform LDPs by using the generalized weak convergence method developed by Salins (Probab. Surv., 16:99-142, 2019). The idea of uniform tail-ends estimates, the pseudo monotone technique and the Arzelà-Ascoli theorem are combined to prove the weak-to-strong continuity of solution operators of the controlled equations in order to overcome many difficulties caused by the noncompactness of Sobolev embeddings on $\mathbb{R}^d$ and the nonlinearity of the fractional $(α,p)$-Laplace operator. The superlinearly growing diffusion term is carefully controlled by using the dissipative drift terms and several algebraic inequalities.
