In search of necessary and sufficient conditions to solve parabolic Anderson model with rough noise

2022/06/06 by Liu, Shuhui, Hu, Yaozhong, Wang, Xiong
#26D15 #60H07 #FOS: Mathematics #Primary 60H15 #Probability (math.PR) #secondary 60H05

paper · doi:10.48550/arxiv.2206.02641

Abstract

This paper attempts to obtain necessary and sufficient conditions to solve the parabolic Anderson model with fractional Gaussian noises: (∂)/(∂ t)u(t,x)=(1)/(2)Δu(t,x)+u(t,x)W(t,x), where W(t,x) is the fractional Brownian field with temporal Hurst parameter H₀∈ [1/2, 1) and spatial Hurst parameters H =(H₁, ⋯, H_d) ∈ (0, 1)^d, and W(t,x)=\frac∂ ^d+1∂ t ∂ x₁ ⋯ ∂ x_dW(t,x). When d=1 and when (H₀,H)∈(\frac 12,1)×(\frac 120,\frac 12) we show that the condition 2H₀+H>5/2 is necessary and sufficient to ensure the existence of a unique solution for the parabolic Anderson Model. When d≥ 2, we find the necessary and sufficient condition on the Hurst parameters so that each chaos of the solution candidate is square integrable.

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