elliptic operator - Project Euclid

J. Math. Soc. Japan

Vol. 43, No. 1, 1991

An ergodic control problem arising from the principal eigenfunction of an elliptic operator By Alain BENSOUSSAN and Hideo

NAGAI(*)

(Received Jan. 9, 1989) (Revised Nov. 27, 1989)

0. Introduction. Let us consider the following second order quasi-linear partial differential equation: (0.1)

$- \frac{1}{2}\Delta v_{\alpha}+H(x, \nabla v_{a})+\alpha v_{\alpha}=0$

, where is a positive with a quadratic growth nonlinear term $H(x, \nabla v.)$ on constant. Such kinds of equations on bounded regions with periodic or Neumann boundary conditions have been studied by several authors (cf. BensoussanFrehse [3], Gimbert [6], Lasry [8], and Lions [9] in connection with ergodic of (0.1) as control problems, where the asymptotic behaviour of the solution is investigated. The problems arise from stochastic control tends to (cf. problem Bensoussan [2]). In those works important steps of the resolution -norms of $av$ and of such problems are to deduce the estimates on the by using the maximum principle and the Bernstein’s method. But similar problems on the whole space have been out of consideration because the method does not work. We may say intuitively that main difficulty to treat such problems on the whole space lies in lack of uniform ergodicity of underlying diffusion processes and it seems to be necessary to employ completely different method. In the present article we specialize the equation (0.1) to the case where $\nabla v_{a}$

$\alpha$

$)$

$v_{a}$

$0$

$\alpha$

$L^{\infty}$

(0.2)

.

$\nabla v_{a}$

$H(x, \nabla v_{\alpha})=\frac{1}{2}|\nabla v_{\alpha}|^{2}-V(x)$

. We notice the relationship between the but treat it on whole Euclidean space equation (0.1) with (0.2) and the eigenvalue problem of a Schr\"odinger operator $R^{n}$

$-(1/2)\Delta+V$ (*)

in

$L^{2}(R^{n})$

:

This author was partially supported by Grant-in-Aid for Scientific Research (No. Science and Culture.

62302006), Ministry of Education,

50

A. BENSOUSSAN and H. NAGAI

(0.3)

$- \frac{1}{2}\Delta\phi+V\phi=\lambda\phi$

.

More precisely, let us take the principal eigenvalue corresponding normalized eigenfunction and set

$\lambda_{1}$

of the operator and the

$\phi(x)$

$w=- \log\phi+\int\phi^{2}\log\phi dx$

then

$w$

,

satisfies the equation

(0.4)

with

$- \frac{1}{2}\Delta w+\frac{1}{2}|\nabla w|^{2}-V(x)+\lambda_{1}=0$

.

$\int w\phi^{2}dx=0$

We start with regarding (0.4) as a Bellman equation of ergodic control type and (0.1) with (0.2) as the corresponding equation of discounted type (cf. \S 1). Our theorems assert that under some conditions on $V(x)\alpha v$ converges to

.

$\lambda_{1}$

, and

$v_{\alpha}- \int v_{\alpha}\phi^{2}dx$

to

$w$

in a suitable function space as

$\alphaarrow 0$

, where

$v_{\alpha}$

is the

positive solution of (0.1) with (0.2) (cf. \S 3). TO study the equation (0.1) with (0.2) we take a transformation. $v_{\alpha}=-\log u_{\alpha}$

and have the equation (0.5)

$- \frac{1}{2}Au_{\alpha}+Vu_{a}=-\alpha u_{\alpha}\log u_{\alpha}$

,

$00$

$G_{\gamma}=(\gamma+H)^{-1},$

is compact.

$0\leqq\lambda_{1}