数理解析研究所講究録
第 1905 巻 2014 年 18-29

18

SPDEs deduced from evolutional models of
two-dimensional Young diagrams
Bin XIE
Department of Mathematical Sciences
Shinshu University
Abstract
It is a survey on the study of limit shapes and fluctuation around relative
limit shapes for both the static and the dynamic two-dimensional Young diagrams. In particular, by studying the non-equilibrium fluctuation problem[6] around their hydrodynamic limits established in [5] for the dynamics of
two-dimensional Young diagrams associated with the uniform and restricted
uniform statistics respectively, two linear stochastic partial diﬀerential equations are deduced to characterize the fluctuation limits. We will also see thatthe results for limit shapes and fluctuation obtained in the static level canbe reinterpreted by that of the evolutional models of two-dimensional Youngdiagrams.

1

Introduction

This note is survey for random Young diagram both in static and dynamic levels.
In particular, by the research on the fluctuation of the dynamics of two dimensional Young diagrams around its hydrodynamic limit [6], we connect the theory of
stochastic partial diﬀerential equations (SPDEs, for short) with the research of large
scale interacting systems, which is an important application of SPDEs.
The stochastic partial diﬀerential equation is one of the most dynamically developing areas of mathematics which lies at the cross section of probability, partial
diﬀerential equations, population biology, mathematical physics, fluid mechanics,
geophysics and finance and others. The research on SPDEs is especially attractive because of its interdisciplinary nature and the enormous richness of current
and potential future applications. Roughly speaking, SPDEs are partial diﬀerential equations added with proper noises, like space-time Gaussian noise and colorednoises. SPDEs have been used to characterize many important models from various
fields, such as stochastic Navier-Stokes equations, stochastic Burgers equations, thecurve of interest rate, filtering. Especially, we point out that SPDEs often appear

19
in the study of stochastic analysis on large scale interacting systems. For example,
Funaki and Olla [4] investigate the fluctuations of the height of ainterface model
on a wall and prove that its limit is the solution of stochastic heat diﬀerential equation with reflection. Please see the lecture note [3] by Funaki for more information
relative tointerface model.
In this note, we will deduce two SPDEs by studying the fluctuation of theevolutional model of two-dimensional Young diagrams, which are initially introduced
by Funaki and Sasada [5] and the main results in published in [6] in 2013. Youngdiagrams are convenient graphical representation of the partition of integers, named
after A. Young from 1900 and are very useful in the theory of group representation.
Before the work of [5], all of the study on random Young diagrams is in the static levelfrom the initial work of Vershik [9] in 1996. However, in the real world, everything isdeveloped, so it is important to consider the evolutional model for Young diagrams.
The following is organized as follows. In the next section, we will give an
explanation of Young diagrams and the review some result on the static model ofYoung diagrams. In Section 3, we will introduce the dynamic model of 2-D Youngdiagrams and consider the hydrodynamic limits of them based on [5] and then in
the last section, we will review the results of fluctuation problems around the thehydrodynamic limits of the dynamic model of 2-D Young diagrams. It is shown thatthe limits are characterized by two SPDEs.
$\nabla\phi$$\nabla\phi$

2

Static Model of $2-D$ Young diagrams

A Young diagram is a diagram relative to partition of positive integers, which is
a combinatorial object useful in group representation theory of the symmetric andgeneral linear groups and named after by A. Young from 1900. In this part, wemainly present some results on the limit shape and fluctuation around it of thestatic Young diagrams associated with the uniform and restricted uniform statistics
respectively.
To make it clear, let us first introduce the partition of positive integers. For
, let
each
be the set of all partitions $p=(p_{i})_{i\in N}$ satisfying $p_{1}\geq p_{2}\geq\cdots\geq$$p_{i}\geq\cdots,$ $p_{i}\in \mathbb{Z}+andn(p)$ $:= \sum_{i\in N}p_{i}=n$. For each partition, the two
dimensional Young diagram associated to it is the diagram by piling up sticks ofheight 1 and side-length , see Figure 2 below.
Vershik in 1996 [9] connected the Young diagrams with the theory of probability.
He introduced some kinds of probability measures on Young diagrams known asstatistical mechanics and then studied the limit shapes of the boundary.of the Youngdiagrams, i.e., the law of large numbers in probability terminology. There are many
examples, which particularly include the uniform and restricted uniform statistics.
These probabilities are known as canonical in Physics and the results relative tothem will be mainly summarized for our -goal.
$n\in \mathbb{N}$$\mathcal{P}_{n}$$p\in \mathcal{P}_{n}$$j$$p_{i}$

20
For each , the uniform statistics is a uniform probability measure denoted byand the restricted uniform statistics is a uniform probability measureon the, which is collection of all partitions withconsidered on the set
denoted by
is a proper subset of
and $q_{i}>q_{i+1}$ if $q_{i}>0$ . It is clear that
The uniform statistics is also called the Bose statistics, which applies only to thoseparticles not limited to single occupancy of the same state. The restricted uniform
statistics is also called the Fermi statistics, that is, the particles in the systems
satisfies the Pauli exclusion principle: two particles can not occupy the same stateat the same time.
To state his results for limit shapes, let us here introduce the height function of
the Young diagrams,$n$$\mu_{U}^{n}$$\mathcal{P}_{n}$$\mathcal{Q}_{n}$$\mu_{R}^{n}$$\mathcal{P}_{n}.$$\mathcal{Q}_{n}$$q=(q_{i})_{i\in N}\in \mathcal{P}_{n}$$\psi_{p}(u):=\sum_{i\in \mathbb{N}}1_{\{u<p_{i}\}}=\#\{i\in \mathbb{N}:p_{i}>u\}, u\geq 0.$

and the scaled height function in the static level$\tilde{\psi}_{p}^{N}(u)=\frac{1}{N}\psi_{p}(Nu) , u\geq 0.$. FurClearly, under such scaling, the area equals to 1, that is,is non-negative, non-increasing and cadlag(rightthermore, it is known that
continuous with left limits).
$\int_{0}^{\infty}\tilde{\psi}_{p}^{N}(u)du=1$$\tilde{\psi}_{p}^{N}(u)$Figure 1: Young Diagram and the height for $n=24=(8,7,4,4,1)$ .
the space of all functions defined on which are
Let us also denote by
is non-negative, non-increasing and cadlag, where $A=\mathbb{R}+orA=\mathbb{R}_{+}^{o}$ $:=$$(0, \infty)$ . We will endow it with the topology of uniform convergence on any compact
subsets,
In [9], Vershik proved the law of large numbers holds, see also [7]. Such kindof results are usually called the equivalence of ensembles in the field of statisticalphysics.
$\mathcal{D}(A)$$\tilde{\psi}_{p}^{N}(u)$$\sim$$\psi$$A$

21
Proposition 2.1. For the uniform statistics, under the probability measureis shown thatconverges to the curve$\mu_{U}^{N^{2}}$, it$\tilde{\psi}_{p}^{N}(u)$$\psi_{U}(u)=-\frac{1}{\alpha}\log(1-e^{-\alpha u})$,in probability in, whereSimilarly, for the restricted uniform statistics, under the probability measure
converges to the curve$\alpha=\pi/\sqrt{6}.$$\mathcal{D}(\mathbb{R}_{+})$$\mu_{R}^{N^{2}},$$\tilde{\psi}_{q}^{N}(u)$$\psi_{R}(u)=\frac{1}{\beta}\log(1+e^{-\beta u})$in probability in$\mathcal{D}(\mathbb{R}_{+}^{o})$

where

.

$\beta=\pi/\sqrt{3}.$the view point of physics, the limit shapes obtained in the above give theexplanation of the limit distribution of the particles’ energy. In addition, the limit
shapesandobtain in the above will be called Vershik’s curves.
Now it is natural to study the fluctuations of the random interfaces around theirlimit shape under some proper scaling. Such problem is an equivalent of the central
limit theorem in probability theory, which is also studied by many authors, see forexample, Vershik et. $a1.[2]$ and Yakubovich [10]. Let$\mathbb{R}om_{\sim}$$\psi_{U}$$\psi_{R}$$\Psi_{N}^{U}(u) :=\sqrt{N}(\tilde{\psi}_{p}^{N}(u)-\psi_{U}(u)) , u>0$$\Psi_{N}^{R}(u) :=\sqrt{N}(\tilde{\psi}_{q}^{N}(u)-\psi_{R}(u)) , u\geq 0,$

where
and
are Vershik’s curve, i.e., the limit shapes obtained in Proposition 2.1. Then the following fluctuations around the Vershik’s curves are establishedin [2, 8, 10], see also [6] for the grandcanonical ensembles.
$\psi_{U}(u)$$\psi_{R}(u)$Proposition 2.2. The fluctuation fields
and
under
, whereandcovariance structures$\Psi_{R}(u)$$\mu_{U}^{N^{2}}$$C_{U}(u, v)= \frac{1}{\alpha}\rho_{U}(u\vee v)$

and$\rho_{U}=-\psi_{U}’,$$\rho_{R}=-\psi_{R}’$$\mu_{R}^{N^{2}}$

,

$u,$

$\Psi_{U}^{N}(u)$$\Psi_{U},$$v>0$ and$\Psi_{R}$weakly converge toare mean Gaussian processes withand$\Psi_{R}^{N}(u)$$\Psi_{U}(u)$$0$$C_{R}(u, v)= \frac{1}{\beta}\rho_{R}(u\vee v)$

are slopes of the Vershik’s$curve\mathcal{S}$,$u,$ $v\geq 0,$, respectively.

These results can be reinterpreted from the dynamic point of view by identifying
the static fluctuation limits with the invariant measures of the limit SPDEs, which
are obtained in Section 4 below.

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Hydrodynamic limits of for the evolutional model of Young diagrams

3

In the above section, we reviewed the convergence of height function of static random
Young diagrams and the fluctuation around its limit shape. From this section, we
wll introduce the dynamics of two-dimensional Young diagrams based on the paper[5] by Funaki and Sasada.
andon
Let us first introduce the grandcanonical ensembles.
Letdefinerespectively, whereon$:= \bigcup_{n}\mathcal{P}_{n}$$\mathcal{P}$$\mu_{U}^{\epsilon}$$\mathcal{Q}$$\mu_{U}^{\epsilon}$$\epsilon\in(0,1)$$:= \bigcup_{n}\mathcal{Q}_{n}$$\mu_{U}^{\epsilon}(p)=\epsilon^{n(p)}/Z_{U}(\epsilon),p\in \mathcal{P}$on$\mathcal{P}$and$\mu_{R}^{\epsilon}(q)=\epsilon^{n(q)}/Z_{R}(\epsilon) , q\in \mathcal{Q}$with the normalizing constants$Z_{U}( \epsilon)=\prod_{k=1}^{\infty}(1-\epsilon^{k})^{-1}$and$Z_{R}( \epsilon)=\prod_{k=1}^{\infty}(1+\epsilon^{k})$.

are two probability measures and the conditionaland
It is known that
respectively identify with canonical ensembles introduced
probabilities on
andin Section 2.
Let us now introduce the dynamics of two-dimensional Young diagrams constructed in [5].
on andFor $0<\epsilon<1$ , the dynamics
andon are introduced as Markov processes on these spaces having generators
acts on functions :, respectively, defined as follows. The operator$\mu_{R}^{\epsilon}(q)$$\mu_{U}^{\epsilon}(p)$$\mathcal{Q}_{n}$$\mathcal{P}_{n}$$p_{t}$$:=p_{t}^{\epsilon}=(p_{i}(t))_{i\in N}$$q_{t}:=q_{t}^{\epsilon}=(q_{i}(t))_{i\in N}$$\mathcal{P}$$L_{\epsilon,U}$$\mathcal{Q}$$f$$L_{\epsilon,U}$$L_{\epsilon,R}$$\mathcal{P}arrow \mathbb{R}$as$L_{\epsilon,U}f(p)= \sum_{i\in N}[\epsilon 1_{\{Pi-1>p_{t}\}}\{f(p^{i,+})-f(p)\}+1_{\{p_{i}>p_{i+1\}}}\{f(p^{i,-})-f(p)\}],$

while the operator$L_{\epsilon,R}$

acts on$f^{r}u$nctions$f$:$\mathcal{Q}arrow \mathbb{R}$as$L_{\epsilon,R}f(q)= \sum_{i\in N}[\epsilon 1_{\{qi-1>q_{i}+1\}}\{f(q^{i,+})-f(q)\}+1_{\{q_{t}>qi+1+1orq_{i}=1\}}\{f(q^{i,-})-f(q)\}],$

where
, and

$i$

$p^{i,\pm}=(p_{j}^{i,\pm})_{j\in N}\in \mathcal{P}$$p_{i}\pm 1$

for $j=i$ andare defined for$q^{i,\pm}\in \mathcal{Q}$andsimilarly for$i\in \mathbb{N}$for $j\neq$by. By convention, we set$p\in \mathcal{P}$$q\in \mathcal{Q}$$p_{j}^{i,\pm}=p_{j}$$p_{0}=q_{0}=\infty.$or
For each initial probability measure on or , we will denote byprocessororon
the law of the trajectory of the Markov
respectively. Then, for every $\epsilon\in(0, i)$ , these Markov processes constructed in$v$$\mathcal{P}$$\mathbb{P}_{\nu}^{N}$$\mathcal{Q}$$p_{t}^{\epsilon}$$q_{t}^{\epsilon}$$\mathcal{D}(\mathbb{R}_{+}^{o}, \mathcal{P})$$\mathbb{Q}_{\nu}^{N}$$\mathcal{D}(\mathbb{R}_{+}, \mathcal{Q})$23
the above have the grandcanonical ensembles
andmeasures, respectively.
In the following, we will choosein each case the averaged size of the Young diagrams under$\mu_{U}^{\epsilon}(p)$$\mu_{R}^{\epsilon}(q)$as their invariant

in such a way thatoris equal to$\epsilon=\epsilon(N)(=\epsilon_{U}(N), \epsilon_{R}(N))$$\mu_{U}^{\epsilon}$$N^{2},$$\mu_{R}^{\epsilon}$$i.e.$$E_{\mu_{U}^{\epsilon(N)}}[n(p)\}=N^{2}$and$E_{\mu_{R}^{\epsilon(N)}}[n(q)]=N^{2}$

Then under the above conditions, the behaviors ofas below.

$\epsilon(N)$as $Narrow\infty$ are formulated

Lemma 3.1.
$\epsilon_{U}(N)=1-\frac{\alpha}{N}+O(\frac{\log N}{N^{2}}) , \alpha=\pi/\sqrt{6}$and$\epsilon_{R}(N)=1-\frac{\beta}{N}+O(\frac{\log N}{N^{2}}) , \beta=\pi/\sqrt{12}.$

Such precise behaviors are very important to study the hydrodynamical limit
and especially, to the fluctuations of the evolutional model of Young diagrams.
Define the corresponding height functions scaled in space and time of the dynamics of the two dimensional.Young diagrams by$\tilde{\psi}_{U}^{N}(t, u)$with$:= \tilde{\psi}_{p_{N^{2}t}^{\epsilon}}^{N}(u)=\frac{1}{N}\psi_{p_{N^{2}t}^{\epsilon}}(Nu)$$and_{\backslash }\tilde{\psi}_{R}^{N}(t, u)$$:= \tilde{\psi}_{q_{N^{2}t}^{\epsilon}}^{N}(u)=\frac{1}{N}\psi_{q_{N^{2}t}^{\epsilon}}(Nu)$,and recalling that
Let us now roughly summarize the results of hydrodynamic limit for dynamics ofYoung diagrams obtained in [5] for the uniform statistics and the restricted statistics
respectively, please see Theorems 2.1 and Theorems 2.2 [5].
$\epsilon=\epsilon(N)$$\tilde{\psi}_{p}^{N}(u)=\frac{1}{N}\sum_{i\in \mathbb{N}}1_{\{Nu<p_{i}\}},$$u\in \mathbb{R}_{+}.$

Theorem 3.2. Let
be a sequence of probability measures onwith the$\delta>0$
property: there exists a function
such that for manyand every
junction with compact support in$(\nu^{N})_{N\in \mathbb{N}}$$\mathcal{P}$$\cdot$$\psi_{0}\in X_{U}$$f$$\mathbb{R}_{+}^{o}$$\lim_{Narrow\infty}\nu^{N}[|\int_{0}^{\infty}f(u)\tilde{\psi}_{p}^{N}(u)du-\int_{0}^{\infty}f(u)\psi_{0}(u)du|>\delta]=0.$Then for each $t>0,$
sense: for any $\delta>0$$\tilde{\psi}_{U}^{N}(t, u)$converges to$\psi_{U}(t, u)$in probability in the following$\lim_{Narrow\infty}\mathbb{P}_{\nu^{N}}^{N}[|\int_{0}^{\infty}f(u)\tilde{\psi}_{U}^{N}(t, u)du-\int_{0}^{\infty}f(u)\psi(t, u)du|>\delta]=$O.

Here $\psi(t, u)$ $:=\psi_{U}(t, u)$ is a unique solution of the following nonlinear partial differential equation $(PDE)$ in :$X_{U}$$\partial_{t}\psi=(\frac{\psi’}{1-\psi})’+\alpha\frac{\psi’}{1-\psi’}, u\in \mathbb{R}_{\mathring{+}}$

,

(3.1)

24
with the initial condition$X_{U}=\{\psi$:$\psi(0, \cdot)=\psi_{U,0}(\cdot)$$\mathbb{R}_{\mathring{+}}arrow \mathbb{R}_{+}^{o};\psi\in C^{1},$, where$\psi’<0,$ $\lim_{u\downarrow 0}\psi(u)=\infty,$ $\lim_{u\uparrow\infty}\psi(u)=0\}.$with theTheorem 3.3. Let
be a sequence of probability measures on$\delta>0$
and everysuch that for many
property: there exists a function
function with compact support in$(\nu^{N})_{N\in N}$$\mathcal{Q}$$\psi_{0}\in X_{R}$$f$$\mathbb{R}_{+}$$\lim_{Narrow\infty}\nu^{N}[|\int_{0}^{\infty}f(u)\tilde{\psi}_{q}^{N}(u)du-\int_{0}^{\infty}f(u)\psi_{0}(u)du|>\delta]=0.$Then for each $t>0,$
sense: for any$\tilde{\psi}_{U}^{N}(t, u)$converges to$\psi_{U}(t, u)$in probability in the following$\delta>0$$\lim_{Narrow\infty}\mathbb{Q}_{\nu^{N}}^{N}[|\int_{0}^{\infty}f(u)\tilde{\psi}_{R}^{N}(t, u)du-\int_{0}^{\infty}f(u)\psi(t, u)du|>\delta]=0.$

is a unique solution of the following nonlinear partial dif$(PDE)$
equation:inferential

Here$\psi(t, u)$$:=\psi_{U}(t, u)$$X_{R}$(3.2)$\partial_{t}\psi=\psi"+\beta\psi’(1+\psi u\in \mathbb{R}+,$

with the initial condition$X_{R}$$:=\{\psi$:$\psi(0, \cdot)=\psi_{R,0}(\cdot)$$\mathbb{R}_{+}arrow \mathbb{R}_{+};\psi\in C^{1},$, where$-1\leq\psi’\leq 0,$$\psi’(0)=-1/2,$ $\lim_{u\uparrow\infty}\psi(u)=0\}.$

It is easy to see that the Vershik’s curves
andobtained in the law oflarge numbers in Section 2 are the unique stationary solution in PDEs (3.1) and(3.2). In this sense, the derivation of the Vershik’s curves is understandable fromthe dynamical point of view.
In the end of this section, we give some ideas to the proof of Theorem 3. andTheorem 3.3. For the grandcanonical uniform and restricted uniform statistics, it
is known that if we consider the gradients of their height functions of the dynamics ofand , then they can be transformed to a weakly asymmetric zero-range
process, respectively a weakly asymmetric simple exclusion process on a set of positive integers with a stochastic reservoir at the boundary site in both processes as
natural time evolutions of the Young diagrams. The weakly asymmetric zero-range
process can be further transformed into a weakly asymmetric simple exclusion process
on the whole integer lattice without any boundary conditions
by rotating the $xy$ -plane around the origin by 45 degrees counterclockwise and projecting the system to the -axis rescaled by. This involves quite a nonlinearity
prove
in
[5].
Anyway,4observed Section of
the hydrodynamic limits above,to
we can turn to study that of a weakly asymmetric simple exclusion process with astochastic reservoir at the boundary site and the discrete type of Hopf-Cole transformation is adapted, which avoid the one-block and two-block estimates. We pointout for the restricted case, the study of the boundary behavior of the transformed
process is vital.
$\psi_{U}$$\psi_{R}$$2^{t}$$p_{t}^{\epsilon}$$q_{t}^{\epsilon}$$0$$\mathbb{Z}$$\overline{\eta}_{t}\in\{0, 1\}^{\mathbb{Z}}$$\sqrt{2}$

$x$

$s$

$0$

25

4

Fluctuation in evolutional Young diagrams andinvariant measures
Similar to the static level, when the hydrodynamic limit for dynamics of Young diagrams, it is natural to consider the fluctuation of the interface around hydrodynamic limit. In this part, we will review some results on the non-equilibrium dynamic
fluctuation for two-dimensional evolutional Young diagrams associated with both
uniform and restricted uniform statistics based on the work [6].
The theory of the equilibrium dynamic fluctuation around the hydrodynamic
limit is well established based on the so-called Boltzmann-Gibbs principle. However,
the results on the non-equilibrium dynamic fluctuations are rather limited and isdue to a special feature of the models.
We will also see that the fluctuations studied in the static level can be reinterpreted from the dynamic point of view by identifying the static fluctuation limits
with the invariant measures of the limit SPDEs.
More precisely, we consider the fluctuations processes
andaround their limits:$\tilde{\psi}_{U}^{N}(t, u)$$\Psi_{U}^{N}(t, u):=\sqrt{N}(\tilde{\psi}_{U}^{N}(t, u)-\psi_{U}(t, u))$and$\Psi_{R}^{N}(t,.u)$$\tilde{\psi}_{R}^{N}(t, u)$$:=\sqrt{N}(\tilde{\psi}_{R}^{N}(t, u)-\psi_{R}(t, u))$,which are elements ofand $D([O, T], D(\mathbb{R}_{+}))$ , respectively. Here let
us first state the result for the case of the restricted uniform statistics [6]. In thefollowing, we will omit all of the assumptions for our main results, please see [6] fordetails.
$D([O, T], D(\mathbb{R}_{\mathring{+}}))$

Theorem 4.1. ($RU$-case) Under some proper assumption, the fluctuation process
converges weakly to
as $Narrow\infty$ on the space $D([O, T], D(\mathbb{R}_{+}))$ for
every $T>0$ . The limit
is in $C([O, T], C(\mathbb{R}_{+}))(a. s.)$ and characterized asa solution of the following SPDE:$\Psi_{R}^{N}(t, u)$$\Psi_{R}(t, u)$$\Psi_{R}(t, u)$$\{\begin{array}{l}\partial_{t}\Psi_{R}(t, u)=\Psi_{R}"(t, u)+\beta(1-2\rho_{R}(t, u))\Psi_{R}’(t, u)+\sqrt{2\rho_{R}(t,u)(1-\rho_{R}(t,u))}\dot{W}(t, u) ,\Psi_{R}’(t, 0)=0,\Psi_{R}(0, u)=\Psi_{R,0}(u) ,\end{array}$where $\rho_{R}(t, u)=-\psi_{R}’(t, u)$ and$\dot{W}(t, u)$Here space-time white noise
noise with covariance$\dot{W}(t, u)$is the space-time white noise on(4.1)$[0, T]\cross \mathbb{R}_{+}.$

means that it is a Gaussian type space-time$\mathbb{E}[\dot{W}(t, u)\dot{W}(s, v)]=\delta(t-\mathcal{S})\delta(u-v)$.

Let us give some explanation for the concept of the solutions to the SPDE (4.1).
We first introduce the concept of weak solutions. Similar to that in PDEs, we say

26
is a solution of the SPDE (4.1) if it is adapted, satisfies $\Psi_{R}\in C([0, T], C(\mathbb{R}_{+}))\cap$$C([O, T], L_{e}^{2}(\mathbb{R}_{+}))(a.s.)$satisfying $f’(t, 0)=0$
and for every
the following holds:$\Psi_{R}(t, u)$$f\in C_{0}^{1,2}([0, T]\cross \mathbb{R}_{+})$$\langle\Psi_{R}(t)$,$f(t)\rangle=\langle\Psi_{R,0},$ $f(0) \rangle+\int_{0}^{t}\langle\Psi_{R}(s)$, $f”(s)-\beta((1-2\rho_{R}(s))f(s))’+\partial_{s}f\rangle ds$$+ \int_{0}^{t}\int_{\mathbb{R}_{+}}f(s, u)\sqrt{2\rho_{R}(s,u)(1-\rho_{R}(\mathcal{S},u))}W$(dsdu) $a.s.$(4.2)

As in the theory of PDEs, we can show that the solution of (4.1) is equivalentis an-valued adapted process and theto its mild form, that is,following holds:$\Psi_{R}(t, u)$$L_{e}^{2}(\mathbb{R}_{+})$$\Psi_{R}(t, u)=\int_{\mathbb{R}_{+}}g(t, u, v)\Psi_{R,0}(v)dv+\int_{0}^{t}\int_{\mathbb{R}_{+}}2\beta g(t-s, u, v)\rho_{R}’(s, v)\Psi_{R}(s, v)dvds$$- \int_{0}^{t}\int_{\mathbb{R}_{+}}\frac{\partial}{\partial v}g(t-s, u, v)\beta(1-2\rho_{R}(s, v))\Psi_{R}(s, v)dvds$$+ \int_{0}^{t}\int_{\mathbb{R}_{+}}g(t-s, u, v)\sqrt{2\rho_{R}(s,v)(1-\rho_{R}(s,v))}W$(dsdv) $a.s.,$

where $g(t, u, v)$ is the fundamental solution to $\partial_{t}\Psi(t, u)=\Psi"(t, u)$ with the homogeneous Neumann boundary condition at , that is$0$$g(t, u, v)= \frac{1}{\sqrt{4\pi t}}\{e^{-\frac{(u-v)^{2}}{4t}}+e^{-\frac{(u+v)^{2}}{4t}}\}, u, v\in \mathbb{R}+\cdot$

It can be proved that the uniqueness of it also holds in the restricted uniform case.
The following, the two SPDEs will obtained by considering the problems offluctuations. The definition for solutions can be introduced similarly as above. So
we will omit the explanation at that time.
Before we state the result for the fluctuation of the uniform case, let us consider
the fluctuation for its rotation process of it. In fact, a natural idea in the -case is
to investigate the fluctuation of the curve, which are obtained
around$\psi_{U}(t)=\{(u,y);y=\psi_{U}(t,u), u\in \mathbb{R}_{+}^{o}\}$by rotating the original curves
andlocated in the first quadrant of the uy-plane by 45 degrees counterclockwise aroundthe origin , respectively, whereis a continuous indented curve obtained
from the graphof the original functionbyfilling all jumps by vertical segments. In particular, this contains a part of -axis:$U$$\check{\psi}_{U}^{N}(t)$$\check{\psi}_{U}(t)$$\tilde{\psi}_{U}^{N}(t)$$O$$\tilde{\psi}_{U}^{N}(t)$$\{(u, y);y=\tilde{\psi}_{U}^{N}(t, u), u\in \mathbb{R}_{+}\}$$\tilde{\psi}_{U}^{N}(t, u)$$y$$\{(0, y);y\geq\tilde{\psi}_{U}^{N}(t,$ $0$

Then it is also interesting to consider the diﬀusive fluctuation$\check{\Psi}_{U}^{N}(t, v):=\sqrt{N}(\check{\psi}_{U}^{N}(t, v)-\check{\psi}_{U}(t, v)) , v\in \mathbb{R},$27which is an element of $D([O, T], C(\mathbb{R}))$ . The fluctuation
defined as above isa natural object to study, since the Young diagrams corresponding to the classbelong to the same class under the reflection with respect to the line $\{y=u\}$ , while
those corresponding to do not have such property in general.
.
Now we can state the result forand then apply it to$\check{\Psi}_{U}^{N}(t)$$\mathcal{P}$$\mathcal{Q}$$\Psi_{U}^{N}(t)$$\check{\Psi}_{U}^{N}(t)$Theorem 4.2. ( -case under rotation) Under some proper assumption, the flucas $Narrow\infty$ on the space
tuation process
converges weakly to$D([O, T], C(\mathbb{R}))$is in
for every $T>$ O. The limit
and characterized as a solution of the following SPDE:$U$$\check{\Psi}_{U}(t, v)$$\check{\Psi}_{U}^{N}(t, v)$$\check{\Psi}_{U}(t, v)$$C([O, T], C(\mathbb{R}))(a.

(4.3)$\{\begin{array}{l}\partial_{t}\check{\Psi}_{U}(t, v)=\frac{1}{2}\check{\Psi}_{U}"(t, v)+\frac{\alpha}{\sqrt{2}}(1-2\rho(t, \sqrt{2}v))\check{\Psi}_{U}’(t, v)+2^{\frac{3}{4}\sqrt{\rho(t,\sqrt{2}v)(1-\rho(t,\sqrt{2}v))}\dot{W}(t,v)},\check{\Psi}_{U}(0, v)=\check{\Psi}_{U,0}(v) ,\end{array}$where$\rho(t, \cdot)$is the solution of the$PDE^{\cdot}$$\{\begin{array}{ll}\partial_{t}\rho(t, v)=\rho"(t, v)+\alpha(\rho(t, v)(1-\rho(t, v t>0, v\in \mathbb{R},\rho(0, v)=\rho_{0}(v) .

and$\dot{W}(t, v)$\mathcal{S}.)$is the space-time white noise on\end{array}$$[0, T]\cross \mathbb{R}.$, we still
andAlthough the directions of the fluctuations are diﬀerent inare able to deduce the next theorem from Theorem 4.2. As pointed out before, thetransformation is nonlinear, so it is important that the convergence in Theorem 4.2
is shown in a function space $D([O, T], C(\mathbb{R}))$ .
$\check{\Psi}_{U}^{N}$$\Psi_{U}^{N}$

Theorem 4.3. ( -case) Under some proper assumption, the fluctuation process
converges weakly to
as $Narrow\infty$ on the space $D([O, T], D(\mathbb{R}_{+}^{o}))$ for
every $T>0$ . The limit
is in $C([O, T], C(\mathbb{R}_{+}^{o}))(a. s.)$ and a solution of thefollowing SPDE:$U$$\Psi_{U}^{N}(t, u)$$\Psi_{U}(t, u)$$\Psi_{U}(t, u)$$\{\begin{array}{l}\partial_{t}\Psi_{U}(t, u)=(\frac{\Psi_{U}’(t,u)}{(1+\rho_{U}(t,u))^{2}})’+\alpha\frac{\Psi_{U}’(t,u)}{(1+\rho_{U}(t,u))^{2}}+\sqrt{\frac{2\rho_{U}(t,u)}{1+\rho_{U}(t,u)}}\dot{W}(t, u) ,\Psi_{U}(0, u)=\Psi_{U,0}(u) ,\end{array}$

(4.4)
where

$\rho_{U}(t, u)=-\psi_{U}’(t, u)$and$\dot{W}(t, u)$is the space-time white noise on$[0, T]\cross \mathbb{R}_{+}^{o}.$Theexistence of SPDE (4.4) is shown by our proof which depends on a nonlinear
transformation. However, because of the nonlinearity, the uniqueness of it is justproved in some small function space, see Section 3[6].

28
Finally, let us study the invariant measures for the SPDEs (4.4) and (4.1).
Roughly speaking, an invariant measure to a Markov process is a probability measuresuch that for any time, the process has the same law . Let
$\mu$

$\mu$

$Q_{U}=- \frac{\partial}{\partial u}\{\frac{1}{\rho_{U}(u)(1+\rho_{U}(u))}\frac{\partial}{\partial u}\}$and$Q_{R}=- \frac{\partial}{\partial u}\{\frac{1}{\rho_{R}(u)(1-\rho_{R}(u))}\frac{\partial}{\partial u}\}.$

Then we have the following results, see Theorem 5.2 and Theorem 5.4 in [6].
Theorem 4.4. The Gaussian measure $N(0, Q_{U}^{-1})$ is the unique invariant measureof the SPDE (4.4), which appeared in Theorem 4.3.
Similarly, the Gaussian measure $N(O, Q_{R}^{-1})$ is the unique invariant measure of
the SPDE (4.1), which appeared in Theorem 4.1.
respectively,
andSince $C_{U}(u, v)$ and $C_{R}(u, v)$ are the Green kernels of
the above theorem gives another proof of fluctuation in the static level for theuniform and the restricted uniform case.
$Q_{U}^{-1}$$Q_{R}^{-1}$

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Department of Mathematical Sciences
Faculty of Science, Shinshu University
3-1-1 Asahi, Matsumoto, Nagano 390-8621, Japan.
-mail: bxie@shinshu-u.ac.jp$:_{E}$