Mild Solution of Semilinear SPDEs with Young Drifts

Jiahao Liang; Shanjian Tang

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arXiv · 2309.06791

Mild Solution of Semilinear SPDEs with Young Drifts

Jiahao Liang·Shanjian Tang

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Abstract

In this paper, we study a semilinear SPDE with a linear Young drift $du_{t}=Lu_{t}dt+f\left(t, u_{t}\right)dt+\left(G_{t}u_{t}+g_{t}\right)d\eta_{t}+h\left(t, u_{t}\right)dW_{t}$, where $L$ is the generator of an analytical semigroup, $\eta$ is an $\alpha$-H\"older continuous path with $\alpha \in \left(1/2, 1\right)$ and $W$ is a Brownian motion. After establishing through two different approaches the Young convolution integrals for stochastic integrands, we introduce the corresponding definition of mild solutions and continuous mild solutions, and give via a fixed-point argument the existence and uniqueness of the (continuous) mild solution under suitable conditions.

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Jiahao Liang, Shanjian Tang. 2023-09-13. Mild Solution of Semilinear SPDEs with Young Drifts. https://arxiv.org/abs/2309.06791

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