Mean-field interaction of Brownian occupation measures, I: uniform tube property of the Coulomb functional

We study the transformed path measure arising from the self-interaction of a three-dimensional Brownian motion via an exponential tilt with the Coulomb energy of the occupation measures of the motion by time $t$. The logarithmic asymptotics of the partition function were identified in the 1980s by Donsker and Varadhan [DV83-P] in terms of a variational formula. Recently [MV14] a new technique for studying the path measure itself was introduced, which allows for proving that the normalized occupation measure asymptotically concentrates around the set of all maximizers of the formula. In the present paper, we show that likewise the Coulomb functional of the occupation measure concentrates around the set of corresponding Coulomb functionals of the maximizers in the uniform topology. This is a decisive step on the way to a rigorous proof of the convergence of the normalized occupation measures towards an explicit mixture of the maximizers, derived in [BKM15]. Our methods rely on deriving H{\"o}lder-continuity of the Coulomb functional of the occupation measure with exponentially small deviation probabilities and invoking the large deviation theory developed in [MV14] to a certain shift-invariant functional of the occupation measures.Comment: To appear in: "Annales de l'Institut Henri Poincare

Vertical technology transfer and the implications of patent protection

Significant amount of vertical technology transfer occurs between developed and developing country firms, yet the literature on intellectual property rights did not pay much attention to this aspect. We show that whether or not the incumbent and the entrant final goods producers are from the same developed country, patent protection in the developing country raises developed-country welfare if (i) patent protection in the developing country deters entry in the final goods market, (ii) the marginal cost difference between the incumbent and the entrant final goods producers is sufficiently small, and (iii) the marginal cost difference between the incumbent and the entrant developing-country firms is sufficiently high.Entry deterrence; Patent; Vertical technology transfer; Welfare

Brownian occupation measures, compactness and large deviations: Pair interaction

Continuing with the study of compactness and large deviations initiated in citeMV14, we turn to the analysis of Gibbs measures defined on two independent Brownian paths in $R^d$ interacting through a mutual self-attraction. This is expressed by the Hamiltonian $intint_R^2d V(x-y) mu(d x)nu(d y)$ with two probability measures $mu$ and $nu$ representing the occupation measures of two independent Brownian motions. Due to the mixed product of two independent measures, the crucial shift-invariance requirement of citeMV14 is slightly lost. However, such a mixed product of measures inspires a compactification of the quotient space of orbits of product measures, which is structurally slightly different from the one introduced in citeMV14. The orbits of the product of independent occupation measures are embedded in such a compactfication and a strong large deviation principle for these objects enables us to prove the desired asymptotic localization properties of the joint behavior of two independent paths under the Gibbs transformation. As a second application, we study the spatially smoothened parabolic Anderson model in $R^d$ with white noise potential and provide a direct computation of the annealed Lyapunov exponents of the smoothened solutions when the smoothing parameter goes to $0$

Brownian occupation measures, compactness and large deviations: Pair interaction

Continuing with the study of compactness and large deviations initiated in \cite{MV14}, we turn to the analysis of Gibbs measures defined on two independent Brownian paths in $\R^d$ interacting through a mutual self-attraction. This is expressed by the Hamiltonian \int\int_{\R^{2d}} V(x-y) \mu(\d x)\nu(\d y) with two probability measures $\mu$ and $\nu$ representing the occupation measures of two independent Brownian motions. Due to the mixed product of two independent measures, the crucial shift-invariance requirement of \cite{MV14} is slightly lost. However, such a mixed product of measures inspires a compactification of the quotient space of orbits of product measures, which is structurally slightly different from the one introduced in \cite{MV14}. The orbits of the product of independent occupation measures are embedded in such a compactfication and a strong large deviation principle for these objects enables us to prove the desired asymptotic localization properties of the joint behavior of two independent paths under the Gibbs transformation. As a second application, we study the spatially smoothened parabolic Anderson model in $\R^d$ with white noise potential and provide a direct computation of the annealed Lyapunov exponents of the smoothened solutions when the smoothing parameter goes to $0$