Local Solution to the Multi-layer KPZ Equation

Abstract

In this article we prove local well-posedness of the system of equations $\partial _t h_{i}= \sum _{j=1}^{i}\partial _x^2 h_{j}+ (\partial _x h_{i})^2 + \xi$ on the circle where $1\le i\le N$ and $\xi$ is a space-time white noise. We attempt to generalize the renormalization procedure which gives the Hopf-Cole solution for the single layer equation and our $h_1$ (solution to the first layer) coincides with this solution. However, we observe that cancellation of logarithmic divergences that occurs at the first layer does not hold at higher layers and develop explicit combinatorial formulae for them.

Appendices

Appendix A: A Short Analysis of $G_i$

Let $\zeta \in \mathbb {R}$. We say that a kernel G defined on $\mathbb {R}^d\setminus \{0\}$ or on a subset thereof is of order $\zeta$ if for all multiindizes k there exists a constant $C>0$ such that

$$\begin{aligned} \sup _{\Vert x\Vert _\mathfrak {s}\le 1} |D^k G(x)|\le C\Vert x\Vert _\mathfrak {s}^{\zeta -|k|_\mathfrak {s}}. \end{aligned}$$

(A.1)

Proposition A.1

The kernel $G_i$ is of order $-1$ for every $i\ge 0$.

Proof

The result is well known for the usual heat kernel, see for instance [6, Lemma 7.4]. If $i=1$, then note that after a partial integration we have the identity

$$\begin{aligned} G_1= \partial _x G *\partial _x G, \end{aligned}$$

(A.2)

and both kernels on the right hand side are of order $-2$. The claim therefore follows from [6, Lemma 10.14]. Note that in that result the kernels are supposed to be compactly supported. However, since the heat kernel decays superexponentially fast at infinity it is possible to adapt this result to the present setting. For $i>1$ we can write $G_i= G_{i-1}* \partial _x^2 G$ so that we can conclude as in the case $i=1$. $\square$

Appendix B: Useful Identity

The identity

$$\begin{aligned} 2G'*\widetilde{G'} = G+\widetilde{G} \end{aligned}$$

(B.1)

was used in [8, Sect. 6] where the tilde is the reflection, i.e. $\widetilde{F}(z):=F(-z)$, and the derivative above is with respect to the spatial variable. Iteratively applying this identity we can get useful identities for convolutions of (derivatives of) $G_m$. For instance we have $4G_0' * \widetilde{G_1'} = 2 \widetilde{G_1} -G_0 -\widetilde{G}_0$, since

$$\begin{aligned} 4 G'* \widetilde{G'} *\widetilde{G''} =2(G + \widetilde{G} )*\widetilde{G''} = 2 \widetilde{G} *\widetilde{G''} -2G'*\widetilde{G'} = 2 \widetilde{G}*\widetilde{G''} -G-\widetilde{G} \end{aligned}$$

where we used integration by parts to shift a derivative $2 G * \widetilde{G''} = -2G'*\widetilde{G'}$ in the second step. When the indices are large this calculation can get more involved, for instance, we have $4G_1' * \widetilde{G_1'} = - G_1 - \widetilde{G_1} + G_0 + \widetilde{G}_0$, because

$$\begin{aligned} 4&G''*G'* \widetilde{G'} *\widetilde{G''} =2 G'' * (G + \widetilde{G} )*\widetilde{G''} =2 G'' * G *\widetilde{G''} +2 G'' * \widetilde{G} *\widetilde{G''} \\&=-2 G'' * G' *\widetilde{G'} -2 G' * \widetilde{G'} *\widetilde{G''} = -G''*(G+\widetilde{G}) - (G+\widetilde{G})*\widetilde{G''} \\&= -G*G'' - \widetilde{G}*\widetilde{G''} - G''*\widetilde{G} - G*\widetilde{G''} = -G*G'' - \widetilde{G}*\widetilde{G''} + 2G'*\widetilde{G'} \\&= -G*G'' - \widetilde{G}*\widetilde{G''} +G+\widetilde{G} \;. \end{aligned}$$

In order to obtain a general set of identities we define kernels $D_{i,j}$, for $(i, j) \in \Lambda {\mathop {=}\limits ^{\text{ def }}}\{0,1,2,\dots \}^{2}$ via

$$\begin{aligned} D_{i,j} {\mathop {=}\limits ^{\text{ def }}}G'_{i} *\widetilde{G'_{j}}. \end{aligned}$$

(B.2)

For $i,j > 0$, one has the recursion relation

$$\begin{aligned} D_{i,j} = - \frac{1}{2} \Big ( D_{i-1,j} + D_{i,j-1} \Big )\;. \end{aligned}$$

Indeed, making use of (B.1), we can write

$$\begin{aligned} \begin{aligned} D_{i,j}&= (\partial _x^2 G)^{*i}*G'*\widetilde{G'}*(\partial _x^2\widetilde{G})^{*j}\\&= \frac{1}{2} \Big ((\partial _x^2 G)^{*i}*G*(\partial _x^2\widetilde{G})^{*j} +(\partial _x^2 G)^{*i}*\widetilde{G}*(\partial _x^2\widetilde{G})^{*j} \Big ) \\&= -\frac{1}{2} \Big ((\partial _x^2 G)^{*i}*G* \partial _x^2\widetilde{G}*(\partial _x^2\widetilde{G})^{*(j-1)} +(\partial _x^2 G)^{*(i-1)}*\partial _x^2 G*\widetilde{G}*(\partial _x^2\widetilde{G})^{*j} \Big ) \end{aligned} \end{aligned}$$

(B.3)

and by shifting a derivative the above recursion relation follows.

When $i = 0, j > 0$ one has the recursion

$$\begin{aligned} D_{0,j} = -\frac{1}{2} D_{0,j-1} +\frac{1}{2} G_{j}. \end{aligned}$$

Since $\widetilde{D_{i,j}} = D_{j,i}$ we get, when $j=0$ and $i > 0$

$$\begin{aligned} D_{i,0} = -\frac{1}{2} D_{i-1,0} +\frac{1}{2} \widetilde{G_{i}}. \end{aligned}$$

Finally, $D_{0,0} = \frac{1}{2}[G_{0} + \widetilde{G_{0}}]$.

We now use these recursions to find formula for $D_{i,j}$. A lattice path is a sequence of nearest neighbor edges (steps) of $\Lambda$ which satisfy the natural adjacency relation.

Let $B{\mathop {=}\limits ^{\text{ def }}}\{(x,y) \in \Lambda , i=0 \text{ or } j=0\}$ be the boundary of the discrete first quadrant. We denote by W(i, j) the set of all lattice paths $\gamma$ which

• Start at (i, j)

• Only move down or to the left

• Terminate at a site of B (note, paths are allowed to travel along B for some time, but they always end at a site of B, they can’t go negative.)

For a lattice path $\gamma$ we denote by $l(\gamma )$ the number of steps in $\gamma$. One then has the following formula

$$\begin{aligned} D_{i,j} = \sum _{\gamma \in W(i,j)} (-2)^{-l(\gamma )} F(\gamma _{\mathrm {end}}) \end{aligned}$$

where $\gamma _{\mathrm {end}}$ is the final site visited by $\gamma$ and F is a map from the sites of B to kernels given as follows: $F(0,0) = \frac{1}{2}(G + \widetilde{G})$ and

$$\begin{aligned} F(0,j) = \frac{1}{2} G_{j} \qquad \text{ and } \qquad F(i,0) = \frac{1}{2} \widetilde{G_{i}}. \end{aligned}$$

For fixed $(i,j) \in \Lambda$ let B(i, j) be the set of sites of B one can reach via walks in W(i, j). We then get

$$\begin{aligned} D_{i,j} = \sum _{(x,y) \in B(i,j)} (-2)^{-(i+j - x - y)} \left( {\begin{array}{c}i + j - x - y\\ i-x\end{array}}\right) F(x,y), \end{aligned}$$

where $\left( {\begin{array}{c}i + j - x - y\\ i-x\end{array}}\right)$ counts the number of paths from (i, j) to (x, y). Equivalently,

$$\begin{aligned} D_{i,j} = - \sum _{k =0}^i (-2)^{-(i+j - k+1)} \left( {\begin{array}{c}i + j - k\\ j\end{array}}\right) \widetilde{G_k} -\sum _{k =0}^j (-2)^{-(i+j - k +1)} \left( {\begin{array}{c}i + j - k\\ i\end{array}}\right) G_k \;. \end{aligned}$$

(B.4)