Motivation and overview

I'm trying to understand the theory of regularity structures and in particular, following this paper I'm looking to the $\Phi^4_d$ model on the $d$-dimensional torus $T^d$ i.e. the SPDE $$ \partial_t \varphi(z)=\Delta \varphi(z) + \varphi^3(z) +\xi(z) $$ where $z=(t,x_1,\cdots, x_d)$ and $\xi(z)$ is the space time white noise.

If we call $K(t,x)$ the heat kernel on the torus i.e. $$ K(t,x):= \frac{1}{(4\pi t)^{\frac{d}{2}}}\sum_{k \in 2\pi \mathbb{Z}^d}e^{-\frac{(x-k)^2}{t}} $$ we can define $$ Z(t,x):=\int_{} K(t-s,x-y) \xi(dz) $$

We define $\rho_\delta$ the smooting kernel on scale $\delta$ i.e. a mollifiers such that $$ |K * \rho_\delta| \lesssim \frac{1}{(\|z\|_s+\delta)^d} $$ where $\|z\|_s:=\sqrt{t}+\sum_{j=1}^d x_i$ is the scaled norm and $*$ is the space-time convolution.

We can (at least formally) define $$ Z_\delta= Z * \rho_\delta $$ At this point it is possible to calculate $Z_\delta^3$ as an iterated Ito integral and we have $$ Z^3_\delta(y)= \int \prod_{j=1}^3 K* \rho_\delta(y-z_j) \xi(dz_1) \xi(dz_2) \xi(dz_3) + 3 \int \left(\int (K* \rho_\delta(y-z) )^2 dz\right)K *\rho_\delta(y-z_1) \xi(dz_1) $$ My doubt is about the second integral.

My problem

The second integral can be rewritten as $3 C_\delta Z_\delta$ where $$ C_\delta= \int_{(0,s) \times T^d} (K* \rho_\delta)^2 dz $$ In the linked paper it is stated that this constant, when the space dimension is $2$ and $\delta \to 0$ diverges logarithmically in delta. I don't understand how it is possible given the bound $$ |K * \rho_\delta| \lesssim \frac{1}{(\|z\|_s+\delta)^2} $$ If instead $d=3$ the same integral should diverge as $\frac{1}{\delta}$.


why is $$ \lim_{\delta \to 0} \int_{(0,s) \times T^2} (K* \rho_\delta)^2 dz \approx \lim_{\delta \to 0} \log(\delta^{-1}) $$



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