Uniform Large Deviations of Mckean-Vlasov Stochastic Fractional $(\alpha,p)$-Laplacian Equations Driven by Superlinear Noise on $\mathbb{R}^d$

Abstract

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 $(\alpha,p)$-Laplacian equations with $\alpha \in (0,1)$ and $p>2$ driven by superlinear multiplicative noise defined on the whole space $\mathbb{R}^d$, where the non-local nonlinear fractional $(\alpha,p)$-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 Arzel\`{a}-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 $(\alpha,p)$-Laplace operator. The superlinearly growing diffusion term is carefully controlled by using the dissipative drift terms and several algebraic inequalities.