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We define in this work a notion of Young differential inclusion $$ dz_t in F(z_t)dx_t, $$ for an $alpha$-Holder control $x$, with $alpha>1/2$, and give an existence result for such a differential system. As a by-product of our proof, we show that a bounded, compact-valued, $gamma$-Holder continuous set-valued map on the interval $[0,1]$ has a selection with finite $p$-variation, for $p>1/gamma$. We also give a notion of solution to the rough differential inclusion $$ dz_t in F(z_t)dt + G(z_t)d{bf X}_t, $$ for an $alpha$-Holder rough path $bf X$ with $alphain left(frac{1}{3},frac{1}{2}right]$, a set-valued map $F$ and a single-valued one form $G$. Then, we prove the existence of a solution to the inclusion when $F$ is bounded and lower semi-continuous with compact values, or upper semi-continuous with compact and convex values.
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Nonlinear Young integrals have been first introduced in [Catellier,Gubinelli, SPA 2016] and provide a natural generalisation of classical Young ones, but also a versatile tool in the pathwise study of regularisation by noise phenomena. We present here a self-contained account of the theory, focusing on wellposedness results for abstract nonlinear Young differential equations, together with some new extensions; convergence of numerical schemes and nonlinear Young PDEs are also treated. Most results are presented for general (possibly infinite dimensional) Banach spaces and without using compactness assumptions, unless explicitly stated.
We give an elementary proof that Davies definition of a solution to a rough differential equation and the notion of solution given by Bailleul in (Flows driven by rough paths) coincide. This provides an alternative point on view on the deep algebraic insights of Cass and Weidner in their work (Tree algebras over topological vector spaces in rough path theory).
This paper concerns existence of right-continuous with bounded variation solutions of a perturbed second-order differential inclusion governed by time and state-dependent maximal monotone operators.
In this article, we propose a new unifying framework for the investigation of multi-agent control problems in the mean-field setting. Our approach is based on a new definition of differential inclusions for continuity equations formulated in the Wasserstein spaces of optimal transport. The latter allows to extend several known results of the classical theory of differential inclusions, and to prove an exact correspondence between solutions of differential inclusions and control systems. We show its appropriateness on an example of leader-follower evacuation problem.
Let $Omega_1,Omega_2$ be functions of homogeneous of degree $0$ and $vecOmega=(Omega_1,Omega_2)in Llog L(mathbb{S}^{n-1})times Llog L(mathbb{S}^{n-1})$. In this paper, we investigate the limiting weak-type behavior for bilinear maximal function $M_{vecOmega}$ and bilinear singular integral $T_{vecOmega}$ associated with rough kernel $vecOmega$. For all $f,gin L^1(mathbb{R}^n)$, we show that $$lim_{lambdato 0^+}lambda |big{ xinmathbb{R}^n:M_{vecOmega}(f_1,f_2)(x)>lambdabig}|^2 = frac{|Omega_1Omega_2|_{L^{1/2}(mathbb{S}^{n-1})}}{omega_{n-1}^2}prodlimits_{i=1}^2| f_i|_{L^1}$$ and $$lim_{lambdato 0^+}lambda|big{ xinmathbb{R}^n:| T_{vecOmega}(f_1,f_2)(x)|>lambdabig}|^{2} = frac{|Omega_1Omega_2|_{L^{1/2}(mathbb{S}^{n-1})}}{n^2}prodlimits_{i=1}^2| f_i|_{L^1}.$$ As consequences, the lower bounds of weak-type norms of $M_{vecOmega}$ and $T_{vecOmega}$ are obtained. These results are new even in the linear case. The corresponding results for rough bilinear fractional maximal function and fractional integral operator are also discussed.
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