Stochastic Analytical Study for Mathematical Physics of Statistical Physics and Quantum Theory

本文研究了稳定白噪声驱动的抛物型随机偏微分方程的时间和静态规律，并研究了随机线性输运方程的逆问题。对于第一个问题，我们利用复插值空间的理论给出了一种方法，并讨论了只具有有限可积性的特定Banach空间中的收敛性。在构造解的过程中，我们成功地证明了非对称独立随机变量在空间中的收敛性。对于后者，我们得到了一个很好的公式，利用高斯白噪声驱动的二次变化过程的非随机性，从随机观测中恢复确定性系数。此外，我们还得到了一些关于加性噪声和乘性噪声问题的公式，这些噪声都是时间和空间的。

The aim of the present project was to consider temporal and statial regularities for stochastic partial differential equations of parabolic type driven by stable white noise, and to consider an inverse problem for stochastic linear transport equations. For the first problem, we made an approach by using a theory of complex interpolation spaces, and we discussed the convergence in a specific Banach space which has restricted integrability only. We succeeded in showing the convergence for non-symmetric independent random variables in the space which appears in constructing the solution. For the latter one, we got a nice formula to recover the deterministic coefficients from a random observation using the non-randomness of the quadratic variation process driven by a Gaussian white noise. Also we got several formulae for the problem both additive and multiplicative noise and both the noise was temporal and spatial.