Abstract
This paper is concerned with the following stochastic heat equations:
where w H is a time independent fractional white noise with Hurst parameter H=(h 1 , h 2 ,..., h d ) , or a time dependent fractional white noise with Hurst parameter H=(h 0 , h 1 ,..., h d ) . Denote |H|=h 1 +h 2 +...+h d . When the noise is time independent, it is shown that if ½ <h i <1 for i=1, 2,..., d and if |H|>d-1 , then the solution is in L 2 and the L 2 -Lyapunov exponent of the solution is estimated. When the noise is time dependent, it is shown that if ½ <h i <1 for i=0, 1,..., d and if |H|>d- 2 /( 2h 0 -1 ) , the solution is in L 2 and the L 2 -Lyapunov exponent of the solution is also estimated. A family of distribution spaces S ρ , ρ∈ \RR , is introduced so that every chaos of an element in S ρ is in L 2 . The Lyapunov exponents in S ρ of the solution are also estimated.
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Accepted 10 October 2000. Online publication 19 February 2001.
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Hu, Y. Heat Equations with Fractional White Noise Potentials. Appl Math Optim 43, 221–243 (2001). https://doi.org/10.1007/s00245-001-0001-2
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DOI: https://doi.org/10.1007/s00245-001-0001-2