What the study found
The study found that the strength |f(t)| of a time-dependent random source component can be uniquely recovered from boundary flux data on a nonempty open subset. The result is proved for both stochastic heat equations and stochastic wave equations.
Why the authors say this matters
The authors suggest that establishing uniqueness for this inverse random source problem is important for understanding stochastic evolution equations. They also report numerical examples that verify the theoretical results.
What the researchers tested
The researchers studied an inverse random source problem for stochastic evolution equations, including stochastic heat and wave equations. The unknown source was modeled as g(x)f(t)W˙(t), where g is the spatial component, f is the unknown time-dependent component, and W˙ is time-dependent Gaussian white noise.
What worked and what didn't
They first established well-posedness of the corresponding stochastic direct problem, meaning the problem has a mathematically sound solution framework. Under suitable regularity conditions, they demonstrated the existence of stochastic strong solutions for both equations and proved uniqueness of the recovery of |f(t)| from boundary flux data.
What to keep in mind
The abstract does not describe the exact boundary measurements beyond a nonempty open subset. It also says the results hold under suitable regularity conditions, but it does not detail those conditions in the available summary.
Key points
- The paper proves uniqueness for recovering the strength |f(t)| of a time-dependent random source component.
- The inverse problem is studied for stochastic heat equations and stochastic wave equations.
- The source is modeled as g(x)f(t)W˙(t), with a spatial part, a time-dependent part, and Gaussian white noise.
- The authors first establish well-posedness for the corresponding stochastic direct problem.
- Numerical examples are reported to verify the theoretical results.
Disclosure
- Research title:
- Uniqueness proven for inverse random source recovery in stochastic equations
- Authors:
- Xu Wang, Guanlin Yang, Zhidong Zhang
- Institutions:
- Chinese Academy of Sciences, Chinese Academy of Sciences, Sun Yat-sen University, University of Chinese Academy of Sciences, University of Chinese Academy of Sciences
- Publication date:
- 2026-07-07
- OpenAlex record:
- View
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