Weekly Bulletin

Pohozaev identities are conservation laws arising from the symmetries of certain operators. Their most classical example, which gives name to them, is an integration by parts identity that exploits the translation and dilation invariances of the Laplacian, with notable applications in nonexistence results for semilinear equations. In this work, we use a novel spectral approach to prove a Pohozaev-type identity for the Spectral Fractional Laplacian, an operator without translation and dilation invariances defined as the spectral power of the Dirichlet Laplacian on a domain, and we deduce nonexistence results for certain semilinear equations. This is a joint work with Itahisa Barrios-Cubas, María del Mar González, and Matteo Bonforte (Universidad Autónoma de Madrid).

Measure-valued Pólya sequences (MVPS) are stochastic processes whose dynamics are governed by generalized Pólya urn schemes with infinitely many colors. Assuming a general reinforcement rule, MVPSs can be viewed as extensions of Blackwell and MacQueen’s Pólya sequence, which characterizes an exchangeable sequence with a Dirichlet process (DP) prior distribution. In this talk, we give a complete account of the class of exchangeable MVPSs in terms of their prior distributions. First, we show that under exchangeability, an MVPS is necessarily balanced and its reinforcement kernel is, after normalization, a regular conditional distribution. As a result, its prior distribution is that of a DP mixture with respect to a latent parameter, which is associated with the conditioning sigma-algebra. Furthermore, we examine the effects of relaxing exchangeability to conditional identity in distribution and find that the two are equivalent for balanced MVPSs. In the second part of the talk, we study Hoeffding decomposability under exchangeability and provide a complete characterization of the class of exchangeable Hoeffding-decomposable sequences. In particular, we show that there exists a random parameter, conditional on which an exchangeable Hoeffding-decomposable sequence is an MVPS. Joint work with Chorbadzhiyska, Y., Sariev, H. and Gerdjikov, S.