数理解析研究所講究録
第 1905 巻 2014 年 1-4

1

Kardar-Parisi-Zhang equationand its approximation
Tadahisa Funaki
Graduate School of Mathematical Sciences,
The University of Tokyo

1$\tau$KPZ equation

The Kardar-Parisi-Zhang (KPZ) equation is a stochastic partial diﬀerential equation (SPDE), which describes the motion of a growing interface with a random
fluctuation. Denoting the height of the interface at time and positionby$h=h(t, x)$ , it has the form
$x\in \mathbb{R}$$t$(1)$\partial_{t}h=\frac{1}{2}\partial_{x}^{2}h+\frac{1}{2}(\partial_{x}h)^{2}+\dot{W}(t, x)$

wheregiven by$\dot{W}(t, x)$,is the space-time Gaussian white noise, whose covariance structure is(2)$E[\dot{W}(t, x)\dot{W}(s, y)]=\delta(x-y)\delta(t-s)$.

We consider the equation (1) in one dimension. This equation is actually ill-posed
because of inconsistency between the nonlinearity and the roughness of the noise.
As is explained in Section 4 for linear stochastic equations under periodic boundary-H\"older continuous in thecondition, the solution $h(t, x)$ is expected to be
space variable for every, so that the nonlinear term
would diverge.
In fact, instead of (1), the renormalized equation$( \frac{1}{2}-\epsilon)$

$x$

(3)

$\epsilon>0$$(\partial_{x}h)^{2}$$\partial_{t}h=\frac{1}{2}\partial_{x}^{2}h+\frac{1}{2}((\partial_{x}h)^{2}-\delta_{x}(x))+\dot{W}(t, x)$,

has the meaning in the following sense: Its Cole-Hopf solution defined as the logarithm of the solution of the linear stochastic heat equation (SHE) with a multiplicative noise:(4),2i.e., $h(t, x)$ $:=\log Z(t, x)$ is a mathematically well-defined object and, by applying
It\^o’s formula for this $h(t, x)$ , we obtain (3) from (4) at least at a heuristic level.
Note that, since $(dW(t, x))^{2}=\delta_{x}(x)dt$ from (2), the term
appears in (3)as an It\^o correction term.
$\partial_{t}Z=\underline{1}_{\partial^{2}Z}+Z\dot{W}(t, x)$$x$-$\frac{1}{2}\delta_{x}(x)$22Approximation of KPZ equation
To give a meaning to (3) more precisely, we need to consider approximation schemesfor (3). First approximation is simple and introduced by(5)$\partial_{t}h=\frac{1}{2}\partial_{x}^{2}h+\frac{1}{2}((\partial_{x}h)^{2}-\xi^{\epsilon})+\dot{W}^{\epsilon}(t, x)$,

is a smeared noise defined by applying a usual conwhere
with, andas
which tends tovolution kernelof (5) is given by theThen, by It\^o’s formula, we easily see that the solutionof the following SHE withof the solution
Cole-Hopf transformthe smeared noise:$\dot{W}^{\epsilon}(t, x)=\dot{W}*\eta^{\epsilon}(t, x)$$\delta_{0}$$\eta^{\epsilon}$$\epsilon\downarrow 0$$\xi^{\epsilon}=\eta_{2}^{\epsilon}(O)$$\eta_{2}^{\epsilon}=\eta^{\epsilon}*\eta^{\epsilon}.$$h=h^{\epsilon}$$h^{\epsilon}=\log Z^{\epsilon}$(6)$Z=Z^{\epsilon}$$\partial_{t}Z=\frac{1}{2}\partial_{x}^{2}Z+Z\dot{W}^{\epsilon}(t, x)$.

. Thus, weconverges to the solution of (4) as
It is also easy to see thatcan show that the solution of (5) converges to the Cole-Hopf solution of the KPZ
equation. M. Hairer [2] has recently succeeded to give a meaning to (3), without
bypassing the Cole-Hopf transform, and proved that the Cole-Hopf solution is theright solution of (3) under the periodic boundary condition.
In [1], we introduced a diﬀerent type of approximation scheme for the KPZ
equation:$Z^{\epsilon}$$Z$$\epsilon\downarrow 0$$h^{\epsilon}$(7)$\partial_{t}h=\frac{1}{2}\partial_{x}^{2}h+\frac{1}{2}((\partial_{x}h)^{2}-\xi^{\epsilon})*\eta_{2}^{\epsilon}+\dot{W}^{\epsilon}(t, x)$.

This type of approximation is appropriate from the view point to identify the in(in our case, thevariant measures, since the applications of a certain operator
convolution operator) to the noise term and the same operator twice to the drift
term usually do not change the structure of the invariant measures; note that theand the convolution operator commute. The Cole-Hopf transsecond derivative
form applied to this equation leads to an SHE with a smeared noise having an extra
complex nonlinear term involving a certain renormalization structure:
$A$

$A$

$\partial_{x}^{2}$(8)$\partial_{t}Z=\frac{1}{2}\partial_{x}^{2}Z+\frac{1}{2}Z\{(\frac{\partial_{x}Z}{Z})^{2}*\eta_{2}^{\epsilon}-(\frac{\partial_{x}Z}{Z})^{2}\}+Z\dot{W}^{\epsilon}(t, x)$.

It is shown that, under the situation that the corresponding tilt process is stationary,
this complex term (the middle term in the right hand side of (8)) can be replaced
by a simple linear term divided by 24 in the limit, so that the limit equation is thelinear SHE:
(9)$\partial_{t}Z=\frac{1}{2}\partial_{x}^{2}Z+\frac{1}{24}Z+Z\dot{W}(t, x)$.

3The constantis specific and frequently appears in KPZ related papers. The
Wiener-It\^o expansion and a similar method for establishing the so-called BoltzmannGibbs principle are eﬀectively used to derive (9) from (8). As a result, it is shownthat the distribution of a two-sided geometric Brownian motion with a height shift
given by Lebesgue measure is invariant under the evolution determined by the SHE(4) on$\frac{1}{24}$$\mathbb{R}.$Multi-component KPZ equation

3

Multi-component KPZ equation is an extension of (1) and written for $h(t, x)=$$(h^{\alpha}(t, x))_{\alpha=1}^{d}\in \mathbb{R}^{d}$

(10)

as
$\partial_{t}h^{\alpha}=\frac{1}{2}\partial_{x}^{2}h^{\alpha}+\frac{1}{2}\Gamma_{\beta\gamma}^{\alpha}\partial_{x}h^{\beta}\partial_{x}h^{\gamma}+\dot{W}^{\alpha}(t, x)$,whereare constants which satisfy the condition
for alland
are independent space-time Gaussian white noises. We introduce
its approximation:$\Gamma_{\beta\gamma}^{\alpha}=\Gamma_{\gamma\beta}^{\alpha}=\Gamma_{\gamma\alpha}^{\beta}$$\Gamma_{\beta\gamma}^{\alpha}$$\alpha,$$\beta,$$\gamma$$\{\dot{W}^{\alpha}(t, x)\}_{\alpha=1}^{d}$(11)$\partial_{t}h^{\alpha}=\frac{1}{2}\partial_{x}^{2}h^{\alpha}+\frac{1}{2}\Gamma_{\beta\gamma}^{\alpha}(\partial_{x}h^{\beta}\partial_{x}h^{\gamma}-\xi^{\epsilon}\delta^{\beta\gamma})*\eta_{2}^{\epsilon}+\dot{W}^{\epsilon,\alpha}(t, x)$,and study its invariant measures.

4Regularity of solutions of linear SPDEs

Finally, we explain the regularity of the solutions of linear SPDEs, and see that itis determined under the balance between the regularizing eﬀect of the diﬀerentialoperators and the roughness of the noise. Let us consider the linear SPDE on a-dimensional torus for simplicity:
$d$

(12)

$\partial_{t}u=-(-\triangle)^{\alpha}u+\dot{W}(t, x) , x\in \mathbb{T}^{d}=[0, 1)^{d},$with$\alpha\in \mathbb{R}$and give a proof of(13), a.s.
$u(t, \cdot)\in\bigcap_{s<\alpha-\frac{d}{2}}H^{s}(\mathbb{T}^{d})$Letsuch thatbehaves as$\{\phi_{k}, \lambda_{k}\}_{k=1}^{\infty}$

be the eigenfunctions and corresponding eigenvalues ofonis a complete orthonormal system of; note thatwith $c>0$ as $karrow\infty$ . Then,$-\triangle$$\{\phi_{k}\}_{k=1}^{\infty}$$L^{2}(\mathbb{T}^{d})$$\lambda_{k}\sim ck^{2/d}$$E[\Vert u(t)\Vert_{H^{s}(T^{d})}^{2}]=E[\Vert(1-\triangle)^{s/2}u(t)\Vert_{L^{2}(\mathbb{T}^{d})}^{2}]$$\mathbb{T}^{d}$$\{\lambda_{k}\}$4$= \sum_{k=1}^{\infty}(1+\lambda_{k})^{s}E[u_{k}(t)^{2}],$where$u_{k}(t)$$:=(u(t), \phi_{k})_{L^{2}(\mathbb{T}^{d})}$. It satisfies the stochastic diﬀerential equation (SDE):$du_{k}(t)=-\lambda_{k}^{\alpha}u_{k}(t)dt+dw_{k}(t)$with independent Brownian motions
solved:$w_{k}(t)$,$:=(W(t), \phi_{k})$. This SDE can be easily$u_{k}(t)=e^{-\lambda_{k}^{\alpha}}tu_{k}(0)+ \int_{0}^{t}e^{-\lambda_{k}^{\alpha}(t-s)}dw_{k}(s)$,and we have that$E[u_{k}(t)^{2}]=e^{-2\lambda_{k}^{\alpha}t}u_{k}^{2}(0)+E[( \int_{0}^{t}e^{-\lambda_{k}^{\alpha}(t-s)}dw_{k}(s))^{2}]$$=e^{-2\lambda_{k}^{\alpha}}tu_{k}^{2}(0)+ \int_{0}^{t}e^{-2\lambda_{k}^{\alpha}(t-s)}ds$$=e^{-2\lambda_{k}^{\alpha}}tu_{k}^{2}(0)+ \frac{1}{2\lambda_{k}^{\alpha}}(1-e^{-2\lambda_{k}^{\alpha}t})$Thus, if$u(0)\in H^{s}(\mathbb{T}^{d})$.

,$E[ \Vert,u(t)\Vert_{H^{s}(\mathbb{T}^{d})}^{2}]\sim\sum_{k=1}^{\infty}\frac{(1+\lambda_{k})^{s}}{2\lambda_{k}^{\alpha}}<\infty\Leftrightarrow s<\alpha-\frac{d}{2}.$This proves (13). In particular, in case $\alpha=1$ and $d=1$ , we see that $u(t, \cdot)\in$(a.s.), and this suggests the ill-posedness of the KPZ equation (1).
$\bigcap_{s<1/2}H^{s}(\mathbb{T})$References[1] T. FUNAKI AND J. QUASTEL, KPZ equation, its renormalization and invariantmeasures, preprint.
[2] M. Hairer, Solving the KPZ equation, Ann. Math, 178 (2013), 559-664.

Graduate School of Mathematical Sciences
The University of Tokyo
Komaba, Tokyo 153-8914
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