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<p>We consider a stochastic equation in ℝ<sup><i>d</i></sup><i>,</i> whose nonlocal part is a convolution operator with nonnegative symbol and the local part is an operator of multiplication by an ergodic field in ℝ<sup><i>d</i></sup><i>.</i> We present the upper and lower bounds for its solutions corresponding to the constant initial data and present an example of the field for which these bounds coalesce as <i>t</i> → ∞ resulting in an asymptotic formula for the logarithm of the solution. We also briefly discuss the spectral aspect of our results.</p>

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On a Continuous Analog of the Parabolic Anderson Model

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  • 发表日期：

    2025年9月1日

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  • Yu. Kondratiev,
  • L. Pastur

摘要

We consider a stochastic equation in ℝd, whose nonlocal part is a convolution operator with nonnegative symbol and the local part is an operator of multiplication by an ergodic field in ℝd. We present the upper and lower bounds for its solutions corresponding to the constant initial data and present an example of the field for which these bounds coalesce as t → ∞ resulting in an asymptotic formula for the logarithm of the solution. We also briefly discuss the spectral aspect of our results.