<p>This paper is mainly concerned with the large deviation principle for a fractional McKean-Vlasov stochastic reaction-diffusion equation defined on <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\displaystyle \mathbb {R}^n\)</EquationSource> <EquationSource Format="MATHML"><math> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> </mstyle> </math></EquationSource> </InlineEquation>, featuring a polynomial drift of arbitrary degree. We first prove the well-posedness of the underlying equation under a dissipative condition, and then show the strong convergence of solutions of the corresponding controlled equation with respect to the weak topology of controls, by employing the idea of uniform tail-ends estimates of solutions in order to circumvent the non-compactness of Sobolev embeddings on unbounded domains. We finally establish the large deviation principle of the fractional McKean-Vlasov equation by the weak convergence method without assuming the time Hölder continuity of the non-autonomous diffusion coefficients.</p>

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Well-Posedness and Large Deviations of Fractional McKean-Vlasov Stochastic Reaction-Diffusion Equations on Unbounded Domains

  • Zhang Chen,
  • Bixiang Wang

摘要

This paper is mainly concerned with the large deviation principle for a fractional McKean-Vlasov stochastic reaction-diffusion equation defined on \(\displaystyle \mathbb {R}^n\) R n , featuring a polynomial drift of arbitrary degree. We first prove the well-posedness of the underlying equation under a dissipative condition, and then show the strong convergence of solutions of the corresponding controlled equation with respect to the weak topology of controls, by employing the idea of uniform tail-ends estimates of solutions in order to circumvent the non-compactness of Sobolev embeddings on unbounded domains. We finally establish the large deviation principle of the fractional McKean-Vlasov equation by the weak convergence method without assuming the time Hölder continuity of the non-autonomous diffusion coefficients.