The Second Variational Formula for Statistical Biharmonic Maps

Recently, the statistical bi-energy functional and its first variational formula were introduced by the author and H. Furuhata. The Maps satisfying the corresponding Euler-Lagrange equation are called statistical biharmonic maps. We present the second variational formula for the statistical bi-energy functional and introduce the notion of stability. If the target statistical manifold is a Hessian manifold, it turns out that the second variational formula can be represented using the Hessian curvature. We provide examples of statistical biharmonic maps into Hessian manifolds that are significant in Hessian and information geometry. We also determine the stability of these statistical biharmonic maps, including the improper affine sphere.