Convergence of stochastic integrals with applications to transport equations and conservation laws with noise
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| Format: | Preprint |
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2024
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| author | Karlsen, Kenneth H. Pang, Peter H. C. |
| author_facet | Karlsen, Kenneth H. Pang, Peter H. C. |
| contents | Convergence of stochastic integrals driven by Wiener processes $W_n$, with $W_n \to W$ almost surely in $C_t$, is crucial in analyzing SPDEs. Our focus is on the convergence of the form $\int_0^T V_n\, \mathrm{d} W_n \to \int_0^T V\, \mathrm{d} W$, where $\{V_n\}$ is bounded in $L^p(Ω\times [0,T];X)$ for a Banach space $X$ and some finite $p > 2$. This is challenging when $V_n$ converges to $V$ weakly in the temporal variable. We supply convergence results to handle stochastic integral limits when strong temporal convergence is lacking. A key tool is a uniform mean $L^1$ time translation estimate on $V_n$, an estimate that is easily verified in many SPDEs. However, this estimate alone does not guarantee strong compactness of $(ω,t)\mapsto V_n(ω,t)$. Our findings, especially pertinent to equations exhibiting singular behavior, are substantiated by establishing several stability results for stochastic transport equations and conservation laws. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_16157 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Convergence of stochastic integrals with applications to transport equations and conservation laws with noise Karlsen, Kenneth H. Pang, Peter H. C. Probability Analysis of PDEs Primary: 60H15, 60G46, Secondary: 60F25 Convergence of stochastic integrals driven by Wiener processes $W_n$, with $W_n \to W$ almost surely in $C_t$, is crucial in analyzing SPDEs. Our focus is on the convergence of the form $\int_0^T V_n\, \mathrm{d} W_n \to \int_0^T V\, \mathrm{d} W$, where $\{V_n\}$ is bounded in $L^p(Ω\times [0,T];X)$ for a Banach space $X$ and some finite $p > 2$. This is challenging when $V_n$ converges to $V$ weakly in the temporal variable. We supply convergence results to handle stochastic integral limits when strong temporal convergence is lacking. A key tool is a uniform mean $L^1$ time translation estimate on $V_n$, an estimate that is easily verified in many SPDEs. However, this estimate alone does not guarantee strong compactness of $(ω,t)\mapsto V_n(ω,t)$. Our findings, especially pertinent to equations exhibiting singular behavior, are substantiated by establishing several stability results for stochastic transport equations and conservation laws. |
| title | Convergence of stochastic integrals with applications to transport equations and conservation laws with noise |
| topic | Probability Analysis of PDEs Primary: 60H15, 60G46, Secondary: 60F25 |
| url | https://arxiv.org/abs/2404.16157 |