Abstract:We consider a natural class of ${\mathbf {R}}^d$-valued one-dimensional stochastic partial differential equations (PDEs) driven by space-time white noise that is formally invariant under the action of the diffeomorphism group on ${\mathbf {R}}^d$. This class contains in particular the Kardar-Parisi-Zhang (KPZ) equation, the multiplicative stochastic heat equation, the additive stochastic heat equation, and rough Burgers-type equations. We exhibit a one-parameter family of solution theories with the following properties:

1. For all stochastic partial differential equations (SPDEs) in our class for which a solution was previously available, every solution in our family coincides with the previously constructed solution, whether that was obtained using Itô calculus (additive and multiplicative stochastic heat equation), rough path theory (rough Burgers-type equations), or the Hopf-Cole transform (KPZ equation).
2. Every solution theory is equivariant under the action of the diffeomorphism group, i.e. identities obtained by formal calculations treating the noise as a smooth function are valid.
3. Every solution theory satisfies an analogue of Itô’s isometry.
4. The counterterms leading to our solution theories vanish at points where the equation agrees to leading order with the additive stochastic heat equation.

In particular, points (2) and (3) show that, surprisingly, our solution theories enjoy properties analogous to those holding for both the Stratonovich and Itô interpretations of stochastic differential equations (SDEs) simultaneously. For the natural noisy perturbation of the harmonic map flow with values in an arbitrary Riemannian manifold, we show that all these solution theories coincide. In particular, this allows us to conjecturally identify the process associated to the Markov extension of the Dirichlet form corresponding to the $L^2$-gradient flow for the Brownian loop measure.

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中文翻译：

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几何随机热方程

摘要：我们考虑由时空白噪声驱动的自然类 ${\mathbf {R}}^d$-valued 一维随机偏微分方程 (PDEs)，它在微分同胚群作用下形式不变${\mathbf {R}}^d$。该类特别包含 Kardar-Parisi-Zhang (KPZ) 方程、乘法随机热方程、加法随机热方程和粗伯格斯型方程。我们展示了具有以下特性的单参数解理论系列：

1. 对于我们类中所有以前有解的随机偏微分方程 (SPDE)，我们族中的每个解都与先前构造的解一致，无论是使用 Itô 演算（加法和乘法随机热方程）获得的，粗略的路径理论（粗略的 Burgers 型方程），或 Hopf-Cole 变换（KPZ 方程）。
2. 每个解论在微分同胚群的作用下都是等变的，即通过将噪声视为平滑函数的形式计算得到的恒等式是有效的。
3. 每个解论都满足 Itô 等距的类似物。
4. 导致我们的求解理论的反项在方程与加性随机热方程的前导顺序一致时消失。

特别地，点（2）和（3）示出的是，令人惊奇地，我们的解决方案的理论享受类似于保持用于随机微分方程（随机微分方程）的斯特拉托诺维奇和Ito解释这两个属性同时。对于具有任意黎曼流形中的值的谐波映射流的自然噪声扰动，我们表明所有这些解理论是一致的。特别是，这使我们能够推测性地确定与对应于布朗环测度的 $L^2$-梯度流的狄利克雷形式的马尔可夫扩展相关联的过程。

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