Geometric stochastic heat equations

TL;DR

This paper develops a unified solution theory for a class of geometric stochastic heat equations, including KPZ and Burgers equations, demonstrating invariance, consistency with previous solutions, and properties analogous to Stratonovich and Itô calculus.

Contribution

It introduces a one-parameter family of solution theories for geometric SPDEs that unify and extend existing approaches, ensuring invariance and consistency.

Findings

Solutions coincide with known solutions where available

Solution theories are equivariant under diffeomorphisms

Solutions satisfy an Itô-like isometry

Abstract

We consider a natural class of $\mathbf{R}^d$-valued one-dimensional stochastic 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 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: - For all 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 It\^o calculus (additive and multiplicative stochastic heat equation), rough path theory (rough Burgers-type equations), or the Hopf-Cole transform (KPZ equation). - 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. - Every solution theory satisfies an analogue of It\^o's isometry. - 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 It\^o interpretations of 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.

Equations

$$\mathcal{E}(\Phi ,\Phi )={\int }_{L_{\mathcal{M}}}{\int }_{S^1}{⟨{\nabla }_x\Phi (u),{\nabla }_x\Phi (u)⟩}_{u(x)}dx\mu (du),$$

$$\mu (du)\propto e^{-H(u)}du,H(u)=\frac{1}{2}{\int }_{S^1}{g}_{u(x)}({\partial }_{x}u,{\partial }_{x}u)dx,(1):heuristic=e01$$

Full text

For all 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. 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. 3.

Every solution theory satisfies an analogue of Itô’s isometry. 4. 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 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. Keywords: Brownian loops, renormalisation, stochastic PDEMSC classification: 60H15

1. 1 Introduction

1 Introduction

One of the main goals of the present article is to build “the” most natural stochastic process taking values in the space of loops in a compact Riemannian manifold $\mathcal{M}$, i.e. a random map $u\colon S^{1}\times{{\mathchoice{\hbox{\bf R}}{\hbox{\bf R}}{\hbox{\scriptsize\bf R}}{\hbox{\tiny\bf R}}}}_{+}\to\mathcal{M}$. For a candidate to qualify for such an admittedly subjective designation, one would like it to be as “simple” as possible, all the while having as many nice properties as possible. In this article, we interpret this as follows.

The process $u$ should be specified by only using the Riemannian structure on $\mathcal{M}$ and no additional data. 2. 2.

The process $u$ should have a purely local specification in the sense that it satisfies the space-time Markov property. 3. 3.

It should be the unique process with these properties among a ‘large’ class of processes.

A natural way of constructing a candidate would be as follows. Let $\mu$ be the measure on $L_{\mathcal{M}}=\mathcal{C}(S^{1},\mathcal{M})$ given by the law of a Brownian loop, i.e. the Markov process with generator the Laplace-Beltrami operator on $\mathcal{M}$, conditioned to return to its starting point at some fixed time (say $1$). We fix its law at time [math] to be the probability measure on $\mathcal{M}$ with density proportional to $p_{1}(x,x)$, which guarantees that $\mu$ is invariant under rotations of the circle $S^{1}$. One can then consider the Dirichlet form $\mathcal{E}$ given for suitably smooth functions $\Phi\colon L_{\mathcal{M}}\to{{\mathchoice{\hbox{\bf R}}{\hbox{\bf R}}{\hbox{\scriptsize\bf R}}{\hbox{\tiny\bf R}}}}$ by

[TABLE]

where $\nabla_{x}\Phi(u)\in T_{u(x)}\mathcal{M}$ denotes the functional gradient of $\Phi$ and ${\langle\cdot,\cdot\rangle}_{u}$ denotes the scalar product in $T_{u}\mathcal{M}$. Unfortunately, while it is possible to show that $\mathcal{E}$ is regular and closable [LoopsRoeckner], so that we can construct a (possibly non-conservative) Markov process from it, the uniqueness of Markov extensions for $\mathcal{E}$ is a hard open problem. Accordingly, point 3 above fails to be verifiable with this approach.

Instead, we will directly make sense of the corresponding stochastic partial differential equation. At the formal (highly non-rigorous!) level, $\mathcal{E}$ is expected to represent the Langevin equation for the measure $\mu$ which, again at a purely heuristic level (but see [Maeda, Driver] for rigorous interpretations of this identity), is given by

[TABLE]