Interpolating the Stochastic Heat and Wave Equations with Time-independent Noise: Solvability and Exact Asymptotics

2021/08/25 by Le Chen, Chen, Le, Nicholas Eisenberg +1
Economics, Econometrics and Finance · Mathematics · #Stochastic processes and financial applications #Financial Risk and Volatility Modeling #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2108.11473

Abstract

In this article, we study a class of stochastic partial differential equations with fractional differential operators subject to some time-independent multiplicative Gaussian noise. We derive sharp conditions, under which a unique global L^p(Ω)-solution exits for all p≥ 2. In this case, we derive exact moment asymptotics following the same strategy in a recent work by Balan et al [1]. In the case when there exits only a local solution, we determine the precise deterministic time, T₂, before which a unique L²(Ω)-solution exits, but after which the series corresponding to the L²(Ω) moment of the solution blows up. By properly choosing the parameters, results in this paper interpolate the known results for both stochastic heat and wave equations.

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