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Quenched asymptotics for a 1-d stochastic heat equation driven by a rough spatial noise. (English) Zbl 1454.60090

Summary: In this note we consider the parabolic Anderson model in one dimension with time-independent fractional noise \(\dot{W}\) in space. We consider the case \(H<\frac{1}{2}\) and get existence and uniqueness of solution. In order to find the quenched asymptotics for the solution we consider its Feynman-Kac representation and explore the asymptotics of the principal eigenvalue for a random operator of the form \(\frac{1}{2}\Delta+\dot{W}\).

MSC:

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60G22 Fractional processes, including fractional Brownian motion
60L20 Rough paths
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