No Arabic abstract
We investigate the kinetic Schrodinger problem, obtained considering Langevin dynamics instead of Brownian motion in Schrodingers thought experiment. Under a quasilinearity assumption we establish exponential entropic turnpike estimates for the corresponding Schrodinger bridges and exponentially fast convergence of the entropic cost to the sum of the marginal entropies in the long-time regime, which provides as a corollary an entropic Talagrand inequality. In order to do so, we profit from recent advances in the understanding of classical Schrodinger bridges and adaptations of Bakry-Emery formalism to the kinetic setting. Our quantitative results are complemented by basic structural results such as dual representation of the entropic cost and the existence of Schrodinger potentials.
In his work about hypocercivity, Villani [18] considers in particular convergence to equilibrium for the kinetic Langevin process. While his convergence results in L 2 are given in a quite general setting, convergence in entropy requires some boundedness condition on the Hessian of the Hamiltonian. We will show here how to get rid of this assumption in the study of the hypocoercive entropic relaxation to equilibrium for the Langevin diffusion. Our method relies on a generalization to entropy of the multipliers method and an adequate functional inequality. As a byproduct, we also give tractable conditions for this functional inequality, which is a particular instance of a weighted logarithmic Sobolev inequality, to hold.
In this paper we prove stable determination of an inverse boundary value problem associated to a magnetic Schrodinger operator assuming that the magnetic and electric potentials are essentially bounded and the magnetic potentials admit a Holder-type modulus of continuity in the sense of $L^2$.
In this paper we consider the Cauchy problem for $2m$-order stochastic partial differential equations of parabolic type in a class of stochastic Hoelder spaces. The Hoelder estimates of solutions and their spatial derivatives up to order $2m$ are obtained, based on which the existence and uniqueness of solution is proved. An interesting finding of this paper is that the regularity of solutions relies on a coercivity condition that differs when $m$ is odd or even: the condition for odd $m$ coincides with the standard parabolicity condition in the literature for higher-order stochastic partial differential equations, while for even $m$ it depends on the integrability index $p$. The sharpness of the new-found coercivity condition is demonstrated by an example.
This paper investigates sufficient conditions for a Feynman-Kac functional up to an exit time to be the generalized viscosity solution of a Dirichlet problem. The key ingredient is to find out the continuity of exit operator under Skorokhod topology, which reveals the intrinsic connection between overfitting Dirichlet boundary and fine topology. As an application, we establish the sub and supersolutions for a class of non-stationary HJB (Hamilton-Jacobi-Bellman) equations with fractional Laplacian operator via Feynman-Kac functionals associated to $alpha$-stable processes, which help verify the solvability of the original HJB equation.
We give a direct proof of the sharp two-sided estimates, recently established in [4,9], for the Dirichlet heat kernel of the fractional Laplacian with gradient perturbation in $C^{1, 1}$ open sets by using Duhamel formula. We also obtain a gradient estimate for the Dirichlet heat kernel. Our assumption on the open set is slightly weaker in that we only require $D$ to be $C^{1,theta}$ for some $thetain (alpha/2, 1]$.