Wave equation with a coloured stable noise

Анотація

Abstract We define a random measure generated by a real anisotropic harmonizable fractional stable field <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>Z</m:mi> <m:mi>H</m:mi> </m:msup> </m:math> {Z^{H}} with stability parameter <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>α</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mn>2</m:mn> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> {\alpha\in(1,2)} and Hurst index <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>H</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mfrac> <m:mn>1</m:mn> <m:mn>2</m:mn> </m:mfrac> <m:mo>,</m:mo> <m:mn>1</m:mn> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> {H\in(\frac{1}{2},1)} and prove that the measure is σ-additive in probability. An integral with respect to this measure is constructed, which enables us to consider a wave equation in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>ℝ</m:mi> <m:mn>3</m:mn> </m:msup> </m:math> {\mathbb{R}^{3}} with a random source generated by <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>Z</m:mi> <m:mi>H</m:mi> </m:msup> </m:math> {Z^{H}} . We show that the solution to this equation, given by Kirchhoff’s formula, has a modification, which is Hölder continuous of any order up to <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mrow> <m:mn>3</m:mn> <m:mo>⁢</m:mo> <m:mi>H</m:mi> </m:mrow> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>∧</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> {(3H-1)\wedge 1} . In the case where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>H</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mfrac> <m:mn>2</m:mn> <m:mn>3</m:mn> </m:mfrac> <m:mo>,</m:mo> <m:mn>1</m:mn> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> {H\in(\frac{2}{3},1)} , we show further that the modification is absolutely continuous.

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