MATHEMATICS

R. Z. KHAS’MINSKII

ON SOME DIFFERENTIAL EQUATIONS ARISING IN THE STUDY OF OSCILLATIONS WITH SMALL RANDOM PERTURBATIONS

(Presented by Academician I. G. Petrovskii on 18 IX 1961)

1. An oscillatory system subject to random perturbations is, under certain assumptions on the nature of the perturbations, a Markov random process of diffusion type. It is of interest to study the limiting distribution of various characteristics associated with this process when the random perturbation tends to zero. This problem is connected with the study of solutions of a certain class of elliptic and parabolic equations of second order with a small parameter (\varepsilon) multiplying the highest derivatives. This class of equations is characterized by the fact that for (\varepsilon=0) (absence of perturbations) it describes motion along closed curves without absorption (the case with absorption is treated in ((^{1}))). If this motion is nonrandom, then (considering for the time being the elliptic case) by a suitable choice of the coordinate system one may restrict oneself to studying the behavior, as (\varepsilon\to 0), of the solution (u_\varepsilon) of the equation

[
\left(L_1(x)+V(x)+\frac{1}{\varepsilon}\frac{\partial}{\partial x_2}\right)u
=
\sum_{i,j=1}^{2} a_{ij}(x_1,x_2)\frac{\partial^2 u}{\partial x_i\partial x_j}
+
]

[
+\sum_{i=1}^{2} b_i(x_1,x_2)\frac{\partial u}{\partial x_i}
+V(x_1,x_2)u+\frac{1}{\varepsilon}\frac{\partial u}{\partial x_2}
=-F(x_1,x_2)
\tag{1}
]

in the circular annulus (K) (\bigl(K={(x_1,x_2): r_1<x_1<r_2}), where ((x_1,x_2)) and ((x_1,x_2+1)) represent one and the same point()), satisfying the conditions

[
u_\varepsilon(r_i,x_2)=f_i(x_2).
\tag{1'}
]

We shall carry out such an investigation for (1), ((1')) and for the more general equation (2) (i.e., for the case where the “principal” motion along closed trajectories is also random), and we shall also consider the parabolic equation.

2. Suppose that the operator (L_1(x)) is elliptic everywhere in (\overline K) (\bigl(\overline K=K\cup\Gamma;\ \Gamma=\Gamma_1\cup\Gamma_2;\ \Gamma_i={x:x_1=r_i}\bigr)). Put

[
L_2(x)=A(x)\frac{\partial^2}{\partial x_2^2}
+B(x)\frac{\partial}{\partial x_2}
\quad (A>a_0>0\ \text{in }K),
]

[
L_2^*u=\frac{\partial^2}{\partial x_2^2}(A(x)\cdot u)
-\frac{\partial}{\partial x_2}(B(x)\cdot u)
]

and assume that the coefficients of the operators (L_1) and (L_2) are twice continuously differentiable in (K), with the first derivatives continuous in (\overline K). Let (V) and (F) be twice continuously differentiable complex-valued functions, with (\operatorname{Re}V(x)=V_1(x)\leq 0), and let (f_i) ((i=1,2)) be continuous complex-valued functions on (\Gamma_i). Denote by (\mu(x_1^0,x_2)=) the density of the invariant measure of the Markov random process on the circle

[
\text{* The equation (1) with a complex function (V) is satisfied, for example, by characteristic functions of certain functionals of the trajectory of the random process associated with the operator }
L_1+\frac{1}{\varepsilon}\frac{\partial}{\partial x_2}\ (^{3}).
]

of the surface (l_{x_1^0}={x: x_1=x_1^0}), corresponding to the operator (L_2) ((\mu) is the unique solution of the equation (L_2^\mu=0) on (l_{x_1^0}), normalized by the condition (\int_0^1 \mu\,dx_2=1); in particular, (\mu=1) if (L_2=L_2^)). For any integrable function (g(x)) put
[
\widetilde g(x_1)=\int_0^1 g(x_1,x_2)\mu(x_1,x_2)\,dx_2 .
]

Theorem 1. The solution (u_\varepsilon(x)) of the equation
[
\left(L_1+V+\frac{1}{\varepsilon}L_2\right)u=-F
\tag{2}
]
in (K), satisfying condition ((1')), converges as (\varepsilon\to0) to the solution (u_0(x_1)) of the equation
[
[\widetilde L_1(x_1)+\widetilde V(x_1)]u_0=-\widetilde F(x_1),
\tag{3}
]
satisfying the conditions
[
u_0(r_i)=
\frac{\displaystyle \int_0^1 a_{11}(r_i,x_2)\mu(r_i,x_2)f_i(x_2)\,dx_2}
{\displaystyle \int_0^1 a_{11}(r_i,x_2)\mu(r_i,x_2)\,dx_2}
=\widehat f_i
\quad (i=1,2).
\tag{3'}
]

Moreover, uniformly in any closed subdomain of (K), the relation
[
u_\varepsilon(x)-u_0(x_1)=O(\sqrt{\varepsilon})
]
holds. In the case (f_i(x_2)=\mathrm{const}), this estimate can be improved: (u_\varepsilon(x)-u_0(x_1)=O(\varepsilon)) uniformly in (K).

Theorem 1 is also valid for the case where the domain under consideration is a disk. (One of the boundary conditions ((3')) must then be replaced by the condition of boundedness at zero of the solution of equation (3).) In addition, Theorem 1 remains valid also in the case (A\equiv0,\ B\ne0). Some probabilistic consequences of Theorem 1 with (A\equiv0) are given in ((^3)).

We now somewhat modify the restrictions adopted above, allowing the coefficients of the operator (L), (V), and (F) also to depend on (t) ((0\le t\le T)), in such a way that (L_1=L_1(x,t)), (V(x,t)), and (F(x,t)) are twice differentiable with respect to all their arguments in the domain (K\times(0,T)).

Theorem 2. The solution (u_\varepsilon(x,t)) of the equation
[
\frac{\partial u}{\partial t}=L_1(x,t)u+V(x,t)u+
\frac{1}{\varepsilon}L_2(x)u+F(x,t)
]
in the domain (K\times(0,T)), satisfying the conditions
[
u_\varepsilon(x_1,x_2,0)=f(x_1,x_2),\qquad
u_\varepsilon(r_i,x_2,t)=f_i(x_2,t)\quad (i=1,2),
]
as (\varepsilon\to0) converges to the solution (u_0(x_1,t)) of the equation
[
\partial u/\partial t=[\widetilde L_1(x_1,t)+\widetilde V(x_1,t)]u+\widetilde F(x_1,t),
]
satisfying the conditions
[
u_0(x_1,0)=\widetilde f(x_1),\qquad
u_0(r_i,t)=\widetilde f_i(t)\quad (i=1,2).
]
Moreover, (u_\varepsilon(x,t)-u_0(x_1,t)=O(\sqrt{\varepsilon})) uniformly in any closed domain contained in (K\times(0,T)).

It is interesting to note that (\lim_{\varepsilon\to0}u_\varepsilon(x,t)), generally speaking, does not exist if (A\equiv0). We also note that a result analogous to Theorem 2 is valid for the solution of the Cauchy problem as well.

1. The proof of Theorem 1 uses the following lemmas.

Lemma 1. Under the assumptions of Theorem 1, the solution (Z(x)) of the equation
[
L_1Z+VZ=-F
]
in the domain (K), satisfying the conditions (Z(r_i,x_2)=f_i(x_2)), admits the estimate
[
|Z|\le
\max_{\substack{x_2,\ i=1,2}} |f_i|
+
C\min_{i=1,2}
\int_{r_1}^{r_2} |r_i-x_1|\,
\max_{x_2}|F(x_1,x_2)|\,dx_1 .
]

Here the constant (C) depends only on the upper and lower bounds of the coefficients (a_{11}) and (b_1) in the domain (K).

Lemma 2. Let (x_2) be the coordinate of a point on the circle (l) (considered modulo 1); (A_1(x_2)>0), (B_1(x_2)) are twice continuously differentiable functions on (l). The equation (Mu=A_1d^2u/dx_2^2+B_1du/dx_2=\Phi(x_2)) has a solution on (l) if and only if

[
\int_0^1 \Phi(y)\mu(y)\,dy=0
]

((\mu(x_2)) is a solution of the equation (M^*\mu=0), satisfying the condition

[
\int_0^1 \mu(y)\,dy=1
]).

Lemma 3. The solution (u(x_1,x_2)) of the elliptic equation (\partial^2u/\partial x_1^2+Mu=0) in the domain (G={x_1>0}\times l), satisfying the condition (u_0(0,x_2)=f(x_2)), admits the estimates:

[
\left|u(x_1,x_2)-\int_0^1 f(y)\mu(y)\,dy\right|\leq c_1e^{-c_2x_1};\qquad
|u'{x_i}|\leq c_3e^{-c_2x_1};\qquad
|u''}|<c_3e^{-c_2x_1
]

(here and below the (c_i) are certain positive constants).

Let (u_\varepsilon(x)) be the solution of equation (2), (1′), and let (v_\varepsilon^{(1)}(x)) be the solution of equation (2), satisfying the conditions (v_\varepsilon^{(1)}(r_i,x_2)=\hat f_i) ((i=1,2)), (v_\varepsilon^{(2)}(x)=u_\varepsilon(x)-v_\varepsilon^{(1)}(x)). We shall prove the following estimates:

[
v_\varepsilon^{(1)}(x)-u_0(x_1)=O(\varepsilon)\quad \text{uniformly in }K;
\tag{4}
]

[
v_\varepsilon^{(2)}(x)=O(\sqrt{\varepsilon})\quad \text{uniformly in any closed subdomain of }K.
\tag{5}
]

Set (\Phi(x)=L_1(x)\cdot u_0+F(x)+V(x)\cdot u_0). Since, by virtue of (3), (\widetilde{\Phi}(x_1)=0), applying Lemma 2 one may assert that the equation (L_2w=\Phi) has a solution on each circle (l_{x_1}), (r_1\leq x_1\leq r_2), where the variable (x_1) enters this equation as a parameter. It is not difficult to show that (w) may be chosen twice continuously differentiable with respect to (x_1), since (A), (B), and (\Phi) possess this property. Now considering the function (\Psi_\varepsilon(x)=v_\varepsilon^{(1)}(x)-u_0(x_1)+\varepsilon w(x)), it is easy to see that (\Psi_\varepsilon(r_i,x_2)=\varepsilon w(r_i,x_2)=O(\varepsilon)) and

[
\left(L_1+V+\frac{1}{\varepsilon}L_2\right)\Psi_\varepsilon
=\varepsilon\,[L_1w+Vw]=O(\varepsilon).
]

Hence, applying Lemma 1, we obtain (4). To prove (5) we construct auxiliary functions (g_i(x_1,x_2,\varepsilon)) ((i=1,2)) so that the function (g_i) has the character of a boundary layer near (\Gamma_i). We shall start here from the function (Z_i(x_1,x_2)), defined in (G) as the solution of the equation

[
L_2(r_i,x_2)Z+a_{11}(r_i,x_2)\frac{\partial^2 Z}{\partial x_1^2}=0;
\tag{6}
]

[
Z(0,x_2)=f_i(x_2)-\hat f_i.
\tag{7}
]

Application of Lemma 4 to (6), (7) makes it possible to establish that

[
|Z_i|<c_1e^{-c_2x_1};\qquad
\left|\frac{\partial Z_i}{\partial x_j}\right|<c_3e^{-c_2x_1};\qquad
\left|\frac{\partial^2 Z_i}{\partial x_j\partial x_k}\right|<c_3e^{-c_2x_1}.
]

Putting now

[
g_i(x_1,x_2,\varepsilon)=Z_i\left(\frac{|x_1-r_i|}{\sqrt{\varepsilon}},x_2\right),
]

we therefore obtain:

[
g_i,\ \frac{\partial g_i}{\partial x_2},\ \frac{\partial^2 g_i}{\partial x_2^2}
=
O\left[\exp\left(-\frac{c_2|x_1-r_i|}{\sqrt{\varepsilon}}\right)\right];
]

[
\frac{\partial g_i}{\partial x_1},\ \frac{\partial^2 g_i}{\partial x_1\partial x_2}
=
O\left[\frac{1}{\sqrt{\varepsilon}}
\exp\left(\frac{c_2|x_1-r_i|}{\sqrt{\varepsilon}}\right)\right];
]

[
\frac{\partial^2 g_i}{\partial x_1^2}
=
O\left[\frac{1}{\varepsilon}
\exp\left(-c_2\frac{|x_1-r_i|}{\sqrt{\varepsilon}}\right)\right].
]

Next, using (6) and these estimates, we have:

[
\begin{aligned}
\left(L_1+V+\frac{1}{\varepsilon}L_2\right)g_i(x,\varepsilon)
&=\left[a_{11}(x_1,x_2)-a_{11}(r_i,x_2)\right]\frac{\partial^2 g_i}{\partial x_1^2}+{}\
&\quad+\frac{1}{\varepsilon}\left[L_2(x_1,x_2)-L_2(r_i,x_2)\right]g_i
+O\left[\frac{1}{\sqrt{\varepsilon}}\exp\left(-\frac{c_2|x_1-r_i|}{\sqrt{\varepsilon}}\right)\right] \
&=O\left[\frac{|x_1-r_i|}{\varepsilon}\exp\left(-\frac{c_2|x_1-r_i|}{\sqrt{\varepsilon}}\right)\right]
+O\left[\frac{1}{\sqrt{\varepsilon}}\exp\left(-\frac{c_2|x_1-r_i|}{\sqrt{\varepsilon}}\right)\right].
\end{aligned}
]

Applying now Lemma 1 to the function

[
v_\varepsilon^{(2)}(x)-\sum_{i=1}^{2} g_i(x,\varepsilon),
]

it is easy to verify that

[
v_\varepsilon^{(2)}(x)-\sum_{i=1}^{2} g_i(x,\varepsilon)=O(\sqrt{\varepsilon})
]

uniformly in (K), whence (5) follows. Theorem 1 follows from (4) and (5).

1. The proof of Theorem 1 in the case (A\equiv 0,\ B\ne0) is analogous. (Instead of Lemma 3 one must use a similar assertion concerning the solution of the equation (\varepsilon^2 u/\partial x_1^2+\partial u/\partial x_2=0) in (G).) The proof of Theorem 2 is constructed according to the same plan. It is only necessary additionally to construct a boundary layer near the plane (t=0). For its construction the following is used:

Lemma 4. The solution of the equation (\partial u/\partial t=Mu) in the domain ({t>0}\times l), satisfying the condition (u(0,x_2)=f(x_2)), admits the estimate

[
\left|u(t,x_2)-\int_{0}^{1} f(y)\mu(y)\,dy\right|