Flow Equation Approach to a Singular SPDE

This presentation outlines the project I developed for my 4th-year colloquium (coloquio di passaggio d’anno) at the Scuola Normale Superiore (SNS) under the supervision of Professor Marco Romito. It was subsequently expanded and presented as the final exam for the Introduction to Stochastic PDEs course at the University of Pisa.

Motivation

Many SPDEs originating from physics are singular, meaning standard tools from probability and analysis cannot be directly applied. An example of this is the Dynamical ${\Phi }^4$ model:

$$({\partial }_t+1-\Delta )\Phi =\xi -\lambda {\Phi }^3.$$

This equation is typically posed in ${\mathbb{R}}_{+}\times {\mathbb{R}}^d$ for dimensions $d\in \{2,3\}$. In the linear case ($\lambda =0$) we can solve the equation and the solution $\Phi$ turns out to be a distribution rather than a classical function. We may expect that for sufficiently small $\lambda$ the equation is just a perturbation of the linear case above, meaning we still expect $\Phi$ to be a distribution. But then the non-linear term ${\Phi }^3$ is classically not well-defined, and so it is not clear what it means for a distribution $\Phi$ to solve the equation.

To understand how we can give meaning to such equations, we focus instead on a simpler variant:

$$(1-\Delta {)}^{\sigma /2}\Phi =\xi +\lambda {\Phi }^3.$$

The Flow Equation Approach

How do we rigorously define a solution to this? We proceed via a two-step regularization and renormalization procedure.

First, we regularize the driving noise $\xi$ using a mollifier. We define a smooth noise approximation:

$${\xi }_{\kappa }={\theta }_{\kappa }\ast \xi \in C^{\infty }({\mathbb{R}}^d)$$

where ${\theta }_{\kappa }\to \delta$ when $\kappa \to 0$.

Using the regularized noise, we attempt to define a sequence of smooth solutions ${\Phi }_k$:

$${\Phi }_k=G\ast F_k[{\Phi }_k]$$

where $G$ is the Green function of the fractional operator $(1-\Delta {)}^{\sigma /2}$, and

$$F_k[\phi ](x)={\xi }_k(x)+\lambda \phi (x{)}^3.$$

However, simply hoping for the convergence of ${\Phi }_{\kappa }$ as $\kappa \to 0$ does not work.

The Main Theorem

To achieve convergence, we must introduce counterterms into our sequence to control the diverging quantities. The modified non-linearity becomes:

$$F_{\kappa }[\phi ](x)={\xi }_k(x)+\lambda \phi (x{)}^3+\sum _{i=2}^N{\lambda }^i{c}_k^{(i)}\phi (x).$$

This procedure is formally denoted as solving

$$(1-\Delta {)}^{\sigma /2}\Phi =\xi +\lambda {\Phi }^3-\infty \Phi .$$

With the proper counterterms in place, we arrive at the main result:

Theorem: Let $\sigma \in (\frac{d}{3},\frac{d}{2}]$. There exists a choice of the counterterms, a random variable ${\lambda }_{\ast }\in [0,1]$, and a distribution $\Phi \in {\mathcal{S}}^{\prime }({\mathbb{R}}^d)$ such that for every $\lambda <{\lambda }_{\ast }$, there exists a ${\Phi }_k$ that solves the renormalized equation. Furthermore, ${\Phi }_k\to \Phi$ almost surely in the ${\mathcal{S}}^{\prime }({\mathbb{R}}^d)$ topology.

We define this limiting distribution $\Phi$ as the rigorous solution to the original singular SPDE.

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Selected Bibliography:

• Duch, P., Lecture notes on flow equation approach to singular stochastic PDEs. arXiv preprint arXiv:2511.07120, 2025.
• Duch, P., Renormalization of singular elliptic stochastic PDEs using flow equation Probability and Mathematical Physics, Vol. 6 (2), 111-138, 2025.