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<p>We consider the stochastic heat equation which includes a fractional power of the Laplacian of order <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\alpha \in (1, 2]\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation> and it is driven by a nonlinear space-time Gaussian white noise. We study two types of power variations for the solution to this equation: the renormalized quadratic variation and the power variation of order <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\frac{2\alpha }{\alpha -1}\)</EquationSource> <EquationSource Format="MATHML"><math> <mfrac> <mrow> <mn>2</mn> <mi>α</mi> </mrow> <mrow> <mi>α</mi> <mo>-</mo> <mn>1</mn> </mrow> </mfrac> </math></EquationSource> </InlineEquation>, both over an equidistant partition of the unit interval. We prove that these two sequences admit nontrivial limits when the mesh of the partition goes to zero. We apply these results to identify certain parameters of the stochastic heat equation.</p>

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Temporal quadratic and higher order variation for the nonlinear stochastic heat equation and applications to parameter estimation

  • 期刊论文

  • 发表日期：

    2025年5月31日

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  • Christian Olivera,
  • Ciprian A. Tudor

摘要

We consider the stochastic heat equation which includes a fractional power of the Laplacian of order \(\alpha \in (1, 2]\) α ∈ ( 1 , 2 ] and it is driven by a nonlinear space-time Gaussian white noise. We study two types of power variations for the solution to this equation: the renormalized quadratic variation and the power variation of order \(\frac{2\alpha }{\alpha -1}\) 2 α α - 1 , both over an equidistant partition of the unit interval. We prove that these two sequences admit nontrivial limits when the mesh of the partition goes to zero. We apply these results to identify certain parameters of the stochastic heat equation.