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<p>In [<CitationRef CitationID="CR8">8</CitationRef>] the authors showed existence and uniqueness of solutions to the nonlinear one-dimensional stochastic heat equation driven by a Gaussian noise that is white in time and rougher than white in space (in particular, its covariance is not a measure). Here we present a simple alternative to derive such results by considering the equations in the analytically weak sense, using either the variational approach or Krylov’s <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(L^p\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation>-theory. Various improvements are obtained as corollaries.</p>

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Analytically weak solutions to stochastic heat equations with spatially rough noise

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年6月14日

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  • Máté Gerencsér

摘要

In [8] the authors showed existence and uniqueness of solutions to the nonlinear one-dimensional stochastic heat equation driven by a Gaussian noise that is white in time and rougher than white in space (in particular, its covariance is not a measure). Here we present a simple alternative to derive such results by considering the equations in the analytically weak sense, using either the variational approach or Krylov’s \(L^p\) L p -theory. Various improvements are obtained as corollaries.