A function$G$in a Bergman space$A$^{$p$}, 0<$p$<∞, in the unit disk$D$is called$A$^{$p$}-inner if |$G$|^{$p$}−1 annihilates all bounded harmonic functions in$D$. Extending a recent result by Hedenmalm for$p$=2, we show (Thm. 2) that the unique compactly-supported solution Φ of the problem $$\Delta \Phi = \chi _D (|G|^p - 1),$$ where χ_{$D$}denotes the characteristic function of$D$and$G$is an arbitrary$A$^{$p$}-inner function, is continuous in$C$, and, moreover, has a vanishing normal derivative in a weak sense on the unit circle. This allows us to extend all of Hedenmalm's results concerning the invariant subspaces in the Bergman space$A$^{2}to a general$A$^{$p$}-setting.

The problem of$subextension$of plurisubharmonic functions is considered. Recently it was shown by Cegrell–Zeriahi, that subextension is always possible for negative plurisubharmonic functions in the energy class ℱ. In this paper we construct, for every hyperconvex domain Ω, a negative plurisubharmonic function in the class ℰ which cannot be subextended. Given any pseudoconvex domain we construct a pluriharmonic function that cannot be subextended.

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We develop a model for the evolution of wealth in a non-conservative economic environment, extending a theory developed earlier by the authors. The model considers a system of rational agents interacting in a game theoretical framework. This evolution drives the dynamic of the agents in both wealth and economic configuration variables. The cost function is chosen to represent a risk averse strategy of each agent. That is, the agent is more likely to interact with the market, the more predictable the market, and therefore the smaller its individual risk. This yields a kinetic equation for an effective single particle agent density with a Nash equilibrium serving as the local thermodynamic equilibrium. We consider a regime of scale separation where the large scale dynamics is given by a hydrodynamic closure with this local equilibrium. A class of generalized collision invariants (GCIs) is developed to overcome the difficulty of the non-conservative property in the hydrodynamic closure derivation of the large scale dynamics for the evolution of wealth distribution. The result is a system of gas dynamics-type equations for the density and average wealth of the agents on large scales. We recover the inverse Gamma distribution, which has been previously considered in the literature, as a local equilibrium for particular choices of the cost function.

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