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<p>In this paper, we consider a boundary-asymptotic value problem for second-order uniformly elliptic equations with possibly unbounded coefficients. Problems of this kind arise in stochastic control settings. By assuming a positive drift, and the finiteness of a hybrid norm of the source, we obtain the existence of solutions, vanishing at infinity, in any unbounded domain, including the whole <i>n</i>-dimensional space (entire solutions). We use a new ABP type estimate, independent on the size and the geometry of domains, due to positive drift, possibly unbounded, and viscosity methods to carry out a converging approximation process. Suitable limitations on the spatial oscillations of second-order terms allow to obtain strong and classical solutions.</p>

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Existence of entire solutions vanishing at infinity by positive drift

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  • 出版模式：

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  • 发表日期：

    2025年2月11日

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  • Antonio Vitolo

摘要

In this paper, we consider a boundary-asymptotic value problem for second-order uniformly elliptic equations with possibly unbounded coefficients. Problems of this kind arise in stochastic control settings. By assuming a positive drift, and the finiteness of a hybrid norm of the source, we obtain the existence of solutions, vanishing at infinity, in any unbounded domain, including the whole n-dimensional space (entire solutions). We use a new ABP type estimate, independent on the size and the geometry of domains, due to positive drift, possibly unbounded, and viscosity methods to carry out a converging approximation process. Suitable limitations on the spatial oscillations of second-order terms allow to obtain strong and classical solutions.