Stochastic Partial Differential Equations with a Linear Potential

许多描述物理世界的偏微分方程，如薛定谔方程，都包含一个势项。我们提出研究具有高斯随机势的随机偏微分方程。这样的方程有很多例子，通常它们是逐案处理的。另一方面，在许多情况下，解可以展开为关于噪声的多个随机积分。有很多工具可以用来研究这样的方程，包括维纳空间分析和费曼-卡茨公式。我们打算用这些工具来研究解的定性性质，如渐近增长和解存在的关键参数。对于某些参数值，我们期望解在Schwartz分布空间中取值。当然，这是一个广泛的程序，许多这样的方程以前已经考虑过了。然而，与白噪声相比，相关高斯噪声的情况鲜为人知。这里，我们指的是特定的现象，这种现象可能只发生于高斯噪声的某些相关函数。同时，随机热方程也受到了越来越多的关注。在物理科学、工程以及越来越多的生物学中，最有用的工具是偏微分方程。求解这个方程使我们对物理情况有了定量的了解，使我们能够做出预测并控制系统。然而，在现实世界中，所有的系统都受到随机噪声的影响，我们的模型也必须包含随机性。随机偏微分方程领域试图解决这种情况。这个领域比偏微分方程要新得多，我们的数学理解也要少得多。这个建议的重点是有潜在项的方程。这样的方程模拟了许多自然现象。在量子力学中，势项表示粒子和作用在系统上的力之间的相互作用。势项也可以出现在流行病传播的方程中，以及在种群生物学的其他情况中。我们将研究随机电位的影响。目标是增加我们对数学工具的理解，以便科学家和工程师能够更熟练地使用它们。

Many partial differential equations describing the physical world, such as the Schrodinger equation, involve a potential term. We propose to study stochastic partial differential equations with a Gaussian random potential. There are many examples of such equations, and usually they are treated on a case-by-case basis. On the other hand, in many cases the solution can be expanded in terms of multiple stochastic integrals with respect to the noise. There are many tools available to study such equations, including Wiener space analysis and the Feynman-Kac formula. We intend to use these tools to study qualitative properties of solutions, such as asymptotic growth, and the critical parameters for existence of solutions. For certain parameter values, we expect solutions taking values in the space of Schwartz distributions. Of course, this is a broad program, and many such equations have been considered before. However, much less is known for the case of correlated Gaussian noise than for white noise. Here, we are referring to specific phenomena, which might occur only for certain correlation functions of the Gaussian noise. Also, the stochastic heat equation has received much more attention than other cases. In the physical sciences, engineering, and increasingly in biology, the most useful tool is a partial differential equation. Solving the equation gives us a quantitative understanding of the physical situation, allowing us to make predictions and to control the system. However, in the real world, all systems are affected by random noise, and our models must also include randomness. The field of stochastic partial differential equations attempts to address this situation. This field is much newer than partial differential equations, and our mathematical understanding is much less. This proposal focuses on equations with a potential term. Such equations model many natural phenomena. In quantum mechanics, potential terms represent the interaction between particles and forces acting on the system. Potential terms can also arise in the equations for the spread of epidemics, and in other situations in population biology. We will study the effect of a random potential. The goal is to increase our understanding of the mathematical tools, so that scientists and engineers can use them more skillfully.