Renormalization of Generalized KPZ Equation

Abstract

We use Renormalization Group to prove local well posedness for a generalized KPZ equation introduced by H. Spohn in the context of stochastic hydrodynamics. The equation requires the addition of counter terms diverging with a cutoff $\epsilon$ as $\epsilon ^{-1}$ and $\log \epsilon ^{-1}$.

Appendices

Appendix A: Proof of Lemma 4

From (37) one has

$$\begin{aligned} {\mathbb E}(\vartheta ^{(N)}_n(t,x),M^{(\alpha )}\vartheta ^{(N)}_n(t,x)) = \bigg (\sum _{\beta =1}^3 M^{(\alpha )}_{\beta \beta } \bigg ) {\mathfrak C}^{(N)}_n(0,0). \end{aligned}$$

Let us split $\mathfrak {C}^{(N)}_n(0,0)$ by isolating the term corresponding to $i=0$ in (35):

$$\begin{aligned} \mathfrak {C}^{(N)}_n(0,0)&= \frac{1}{2^{7/2} \sqrt{\pi }}\int _0^{\infty } \frac{\chi (s)^2 - \chi '(L^{2(N-n)} s)^2}{s^{3/2}}\,ds + R. \end{aligned}$$

(138)

where to stress the cutoff dependence we wrote this with the lower cutoff $\chi '$.

The remainder is easily bounded by

$$\begin{aligned} R \le C e^{-cL^{2n}} \end{aligned}$$

and its change with cutoff by

$$\begin{aligned} |R-R' |&\le C e^{-cL^{2N}}\Vert \chi - \chi '\Vert _{\infty }. \end{aligned}$$

For the main term in (138) we define

$$\begin{aligned} \rho _\chi =\int _0^{\infty } \frac{1 - \chi ( s)^2}{s^{3/2}}\,ds . \end{aligned}$$

Then

$$\begin{aligned}&\int _0^{\infty } \frac{\chi (s)^2 - \chi '(L^{2(N-n)} s)^2}{ s^{3/2}}\,ds =L^{N-n}\rho _{\chi '}-\rho _\chi . \end{aligned}$$

Setting $\delta _n^{(N)}=\sum _{\beta =1}^3 M^{(\alpha )}_{\beta \beta }(R-\rho _\chi )$ the claim follows. $\square$

Appendix B: Proof of Lemma 14

• (a) We have:

  $$\begin{aligned} {\mathfrak C}'_n(t, x)=-\Delta \int _0^{\infty }H_n(t+2s,x)\chi _{N-n}(t+s)\chi '_{N-n}(s)ds \end{aligned}$$

  (139)

  where $\chi '_{N-n}(t)=\chi (t)-\chi (L^{2(N-n)}t)$. Therefore, since $\chi _{N-n}(t+s)\chi '_{N-n}(s) \le \mathbf {1}_{[0,2]}(s)\mathbf {1}_{[0,2]}(t)$, one has

  $$\begin{aligned} | {\mathfrak C}'_n(t, x)| \le C \mathbf {1}_{[0,2]}(t) \sum _{j \in {\mathbb Z}} \ell (t,x+jL^n) \end{aligned}$$

  (140)

  where

  $$\begin{aligned} \ell (t,x+jL^n)&\le C \int _0^2 ds (t+2s)^{-\frac{3}{2}} e^{-\frac{x^2}{4(t+2s)}}[1 + x^2 (t + 2s)^{-1}] \nonumber \\&\le C e^{-cx^2}(x^2+t)^{-{_1\over ^2}}[1 + x^2(x^2+t)^{-1}]\mathbf {1}_{[0,2]}(t)\nonumber \\&\quad + e^{-cx^2/t}t^{-\frac{3}{2}}[1+x^2 t^{-1}]\mathbf {1}_{[2,\infty )}(t). \end{aligned}$$

  (141)

  Combining (140) with (141) one gets

  $$\begin{aligned} {\mathcal C}_n(\tau ,x)\le C e^{-cx^2}(x^2+t)^{-{_1\over ^2}}[1 + x^2(x^2+t)^{-1}]\mathbf {1}_{[0,2]}(t) \in L^p({\mathbb R}\times {\mathbb T}_n) \end{aligned}$$

  (142)

  for $p<3$. To show (114), note that

  $$\begin{aligned} \chi _{N-n}( t+s)|\chi _{\epsilon }(s)-\chi '_{N-n}(s)|\le \mathbf {1}_{[\epsilon ^2,2\epsilon ^2]}(s) \mathbf {1}_{[0,2]}(t) \Vert \chi -\chi '\Vert _\infty \end{aligned}$$

  where $\epsilon = L^{-(N-n)}$. Hence

  $$\begin{aligned} \delta {\mathcal C}_n(t,x)\le C \sum _{j \in {\mathbb Z}}\ell _{N-n}(t,x + jL^n)\mathbf {1}_{[0,2]}(t)\Vert \chi -\chi '\Vert _\infty \end{aligned}$$

  (143)

  where

  $$\begin{aligned} \ell _{M}(t,x):=\int _0^{2L^{-2M}} (t+2s)^{-\frac{3}{2}} e^{-\frac{x^2}{4(t+2s)}}[1 + x^2 (t + 2s)^{-1}]ds=L^{M} \ell _{0}(L^{2M}t,L^{M}x). \end{aligned}$$

  (144)

  Hence using (141) we have

  $$\begin{aligned} \Vert \ell _{M}(t,x)\mathbf {1}_{[0,2]}(t)\Vert _p^p&=L^{-(3-p)M}\Vert \ell _{0}(t,x)\mathbf {1}_{[0,2L^{2M}]}(t)\Vert _p^p \nonumber \\&\le C L^{-(3-p)M}\bigg (1+\int _2^{2L^{2M}} t^{\frac{3}{2}(1-p)} dt \bigg )\le C L^{-\lambda M} \end{aligned}$$

  (145)

  with $\lambda >0$ for $p<3$.

• (b) The claim follows with the same strategy employed in item (a). $\square$

Appendix C: Proof of Lemma 15

First of all, we note that we can replace $H_n$ (the heat kernel on ${\mathbb T}_n$) by H (the heat kernel on ${\mathbb R}$) in ${{\mathfrak C}_n}$ and ${\mathcal Y}_n$. Indeed, letting $\tilde{\mathcal K}$ denote the kernels ${{\mathfrak C}_n}$, ${\mathcal Y}_n$ and $J_n$ built out of H we get

$$\begin{aligned} |\tilde{\mathcal K}(z)-{\mathcal K}(z)|\le Ce^{-|x|}\mathbf {1}_{[0,2]}(t). \end{aligned}$$

Therefore, in the following proof we will consider kernels built with H and drop the tildes.

Let $\epsilon = L^{-(N-n)}$ and $\chi _{\epsilon }=\chi _{N-n}$. We will indicate the scale dependence of the kernels by $\epsilon$ instead of n, i.e. ${\mathfrak C}_n = {\mathfrak C}_{\epsilon }$ and so on. We work in Fourier space in the x variable:

$$\begin{aligned} \widehat{\mathfrak C}_{\epsilon }(t,p)&= p^2e^{-tp^2} \int _0^{\infty } ds \, e^{-2sp^2} \chi _{\epsilon }(t + s)\chi _{\epsilon }( s) = e^{-tp^2}h_\epsilon (t,\sqrt{t} p) \end{aligned}$$

where $h_{\epsilon }$ is defined in (132) and it is uniformly bounded on ${\mathbb R}_+\times {\mathbb R}$. For $Y_{\epsilon }$ we have

$$\begin{aligned} \widehat{Y}_{\epsilon }(t,p)&= {\mathrm i}p \, e^{-tp^2} \chi _{\epsilon }(t ). \end{aligned}$$

Thus

$$\begin{aligned} \widehat{J}_{\epsilon }(t,p)&={\mathrm i}\int _{\mathbb R}dq (p+q)e^{-t((p+q)^2+q^2)}h_\epsilon (t,\sqrt{t} q) \chi _{\epsilon }(t )= \frac{{\mathrm i}p}{\sqrt{t}}\widehat{\mathcal W}_\epsilon (t,\sqrt{t}p) \end{aligned}$$

(146)

where

$$\begin{aligned} \widehat{\mathcal W}_\epsilon (t,r)=\int _{\mathbb R}dq(1+q/r)e^{-((r+q)^2+q^2)}h_\epsilon ( t,q) \chi _{\epsilon }(t ). \end{aligned}$$

$\widehat{\mathcal W}_\epsilon$ is an entire function in r with

$$\begin{aligned} |\widehat{\mathcal W}_\epsilon (t,r)|\le C e^{-c(\mathrm {Re}\, {r})^2} \end{aligned}$$

(147)

if $|\mathrm {Im}\,{r}|\le 1$ (we used $h( t,q)=h(t,-q)$). Hence in particular the inverse Fourier transform ${\mathcal W}_\epsilon (t,x)$ is in $L^1({\mathbb R})$ uniformly in t. We end up with the claim with

$$\begin{aligned} W_\epsilon (z)=\frac{1}{{t}}{\mathcal W}_\epsilon (t,x/\sqrt{t}). \end{aligned}$$

• (b) It suffices to study $A_\epsilon =Y_\epsilon *\partial _x W_\epsilon$. We get

  $$\begin{aligned} \widehat{A}_{\epsilon }(t,p)&=-p^2\int _0^t e^{-(t-s)p^2} \chi _{\epsilon }(t-s )\frac{1}{\sqrt{s}}\widehat{W}_\epsilon (s,\sqrt{s}p)ds=\frac{1}{\sqrt{t}}\hat{a}_\epsilon (t,\sqrt{t}p) \end{aligned}$$

  with

  $$\begin{aligned} \hat{a}_\epsilon (t,p)=-p^2\int _0^1e^{-(1-\sigma )p^2} \chi _{\epsilon }((1-\sigma )t )\frac{1}{\sqrt{\sigma }}\widehat{W}_\epsilon (\sigma t,\sqrt{\sigma }p)d\sigma . \end{aligned}$$

  $\hat{a}_\epsilon$ is entire satisfying (147) and the claim follows.

• (c) These claims follow from

  $$\begin{aligned} |\widehat{\mathcal W}_\epsilon (t,r)-\widehat{\mathcal W}'_\epsilon (t,r)|\le C e^{-c(\mathrm {Re}\, {r})^2}\mathbf {1}_{[{_1\over ^2}\epsilon ^2,2\epsilon ^2]}(t). \end{aligned}$$

  (148)

• (d) Let $B_{\epsilon }=\partial _x {\mathfrak C}_{\epsilon }^2 = 2{\mathfrak C}_{\epsilon }\partial _x{\mathfrak C}_{\epsilon }$. Then

  $$\begin{aligned} \widehat{B}_{\epsilon }(t,p)&=2{\mathrm i}\int _{\mathbb R}dq (p+q)e^{-t((p+q)^2+q^2)}h_\epsilon (t,\sqrt{t} p)h_\epsilon (t,\sqrt{t} q). \end{aligned}$$

  Comparing with (146) and noting that

  $$\begin{aligned} |2h_\epsilon ( t,\sqrt{t} p)-\chi _{\epsilon }(t )|\le C(\mathbf {1}_{[{_1\over ^2}\epsilon ^2,2\epsilon ^2]}(t)+\mathbf {1}_{[{_1\over ^2},2]}(t)) \end{aligned}$$

  we get

  $$\begin{aligned} |J_{\epsilon }(z)-\partial _x {\mathfrak C}_{\epsilon }(z)^2|\le C \left( \epsilon ^{-3}e^{-c|x|/\epsilon }\mathbf {1}_{[{_1\over ^2}\epsilon ^2,2\epsilon ^2]}(t)+e^{-c|x|}\mathbf {1}_{[{_1\over ^2},2]}(t)\right) . \end{aligned}$$

  In the same way we get

  $$\begin{aligned} |Y_{\epsilon }(z)-2\partial _x {\mathfrak C}_{\epsilon }(z)|\le C \left( \epsilon ^{-2}e^{-c|x|/\epsilon }\mathbf {1}_{[{_1\over ^2}\epsilon ^2,2\epsilon ^2]}(t)+e^{-c|x|}\mathbf {1}_{[{_1\over ^2},2]}(t)\right) . \end{aligned}$$

  $\square$