We develop a martingale approach for a class of singular stochastic PDEs of Burgers type (including fractional and multi-component Burgers equations) by constructing a domain for their infinitesimal generators. It was known that the domain must have trivial intersection with the usual cylinder test functions, and to overcome this difficulty we import some ideas from paracontrolled distributions to an infinite dimensional setting in order to construct a domain of controlled functions. Using the new domain, we are able to prove existence and uniqueness for the Kolmogorov backward equation and the martingale problem. We also extend the uniqueness result for "energy solutions" of the stochastic Burgers equation of [GP18a] to a wider class of equations.

中文翻译：

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随机伯格斯方程的无穷小生成器

我们通过为它们的无穷小生成器构造一个域，为一类 Burgers 类型的奇异随机 PDE（包括分数和多分量 Burgers 方程）开发了一种鞅方法。众所周知，该域必须与通常的圆柱测试函数有微不足道的交集，为了克服这个困难，我们将一些想法从平行控制分布导入到无限维设置中，以构建受控函数域。使用新域，我们能够证明 Kolmogorov 后向方程和鞅问题的存在性和唯一性。我们还将 [GP18a] 的随机 Burgers 方程的“能量解”的唯一性结果扩展到更广泛的方程类。