Mathematics

R. Z. Khasminskii

On an Estimate for the Solution of a Parabolic Equation and Some of Its Applications

(Presented by Academician A. N. Kolmogorov on 23 XI 1961)

In the present note a new estimate is given for the solution of a parabolic equation. This estimate is then applied to the investigation of solutions of parabolic and elliptic equations with a small parameter at the higher derivatives, and also to the study of the limiting behavior of the invariant measure of a Markov process with small diffusion.

1. Let \(G\) be a finitely connected domain with differentiable boundary \(\Gamma_G\) in Euclidean space \(E_N\). Denote
\[ D_T=\{(x,s):x\in G,\ s\in(0,T)\} \]
and
\[ \Delta_T=\{(x,s):x\in E_N,\ s\in(0,T)\}. \]
We shall also put
\[ \Gamma=\{(x,s):(x\in\Gamma_G,\ s\in(0,T))\cup[x\in G,\ s=0]\}, \]
when the domain \(D_T\) is considered, and
\[ \Gamma=\{(x,s):s=0\}, \]
when the domain \(\Delta_T\) is considered. Suppose that the function \(u(x,s)\) satisfies the differential equation
\[ \frac{\partial u}{\partial s} = \sum_{i,j=1}^{N} a_{ij}(x,s)\frac{\partial^2 u}{\partial x_i\partial x_j} + \sum_{i=1}^{N} b_i(x,s)\frac{\partial u}{\partial x_i} + c(x,s)u+F(x,s) = L(x,s)u+F(x,s) \tag{1} \]
in one of the domains \(D_T\) or \(\Delta_T\), and
\[ \lim_{(x,s)\to z\in\Gamma} u(x,s)=0. \tag{1'} \]

We subject the coefficients of equation (1) to the following requirements:

\(1^\circ.\) The eigenvalues of the matrix \(((a_{ij}))\) are nonnegative for all \((x,s)\in D_T\) (or \(\Delta_T\)).

\(2^\circ.\) The coefficients \(a_{ij}, b_i, c\), and \(F\) are continuous in \((x,s)\) and bounded in absolute value by a constant \(C\) for all \((x,s)\in D_T\) (or \(\Delta_T\)); moreover, the functions \(a_{ij}\) and \(b_i\) satisfy a Hölder condition.

\(3^\circ.\) For some function \(\varphi(z)\), continuous for \(z\ge 0\), \(\varphi(0)=0\), everywhere in \(D_T\) (or \(\Delta_T\)) the inequality
\[ |F(x,t)-F(y,t)|\le \varphi(|x-y|) \]
holds.

\(4^\circ.\) For all \((x,s)\in D_T\) (or \(\Delta_T\)) the inequality
\[ \left|\int_0^s F(x,u)\,du\right|<A \]
holds.

Theorem 1. Let conditions \(1^\circ\)—\(4^\circ\) be fulfilled. Then, for arbitrary \(\varepsilon>0,\ \eta>0\), the estimate
\[ |u(x,s)|\le K\left[\eta+\varphi(\varepsilon)+A\left(\frac{1}{\varepsilon^{5}\eta^{5/4}}+1\right)\right]. \tag{2} \]
is valid. Here the constant \(K\) depends only on the constants \(T,N\), and \(C\).

Remark. In the case of the domain \(\Delta_T\), instead of uniform boundedness of the coefficients one may require that they grow no faster than \(C|x|\). Then estimate (2) will be valid only in each compact set, and \(K\) will depend on \(T,N,C\), and this compact set.

The proof of Theorem 1 uses the probabilistic representation of the solution of problem (1), (1′) and the following estimate, obtained from a well-known theorem of A. N. Kolmogorov ((¹), Lemma 2.2), for the modulus of continuity of the trajectories of a random process connected with the operator \(L(x,s)\).

2. Estimate (2) can be applied to the study of the dependence of the solution of equation (1) on parameters entering into the equation. We shall carry out

all further considerations only for the Cauchy problem (for the domain \(\Delta_T\)), although analogous results are valid also for the mixed problem, and for the problem without initial conditions.

Suppose that all coefficients of equations (1) depend on a parameter \(\lambda\) \((\lambda\in\Lambda)\). We shall now write equation (1) in the form \(\partial u/\partial s=L_\lambda u+F(x,s,\lambda)\). Let \(\lambda_0\) be a limit point of the set \(\Lambda\) \((\lambda_0\in\Lambda)\).

At this point we shall assume that the following conditions are satisfied:

\(A_1\). The matrix \(\bigl((a_{ij}(x,s,\lambda))\bigr)\) has nonnegative eigenvalues for all \((x,s)\in\Delta_T,\ \lambda\in\Lambda\). All coefficients of equation (1) are continuous and bounded in the domain \(\Delta_T\times\Lambda\) by a constant \(C\), and the conditions ensuring the existence of a solution of the Cauchy problem for equation (1) for each \(\lambda\in\Lambda\) are satisfied.

\(A_2\). The coefficients of equation (1) are continuous in \(x\) uniformly with respect to \((x,s,\lambda)\in E_N\times(0,T)\times\Lambda\).

\(A_3\). Uniformly in \((x,s)\in\Delta_T\), the limits
\[ \lim_{\lambda\to\lambda_0}\int_0^s a_{ij}(x,u,\lambda)\,du = \int_0^s a_{ij}(x,u,\lambda_0)\,du \]
exist, and there are analogous limits for \(b_i,\ c\), and \(F\).

Theorem 2. If conditions \(A_1\)—\(A_3\) are satisfied, then the solution \(u_\lambda(x,s)\) of equation (1) in \(\Delta_T\), satisfying the condition \(u_\lambda(x,0)=f(x)\), converges uniformly in \(\Delta_T\) to the function \(u_{\lambda_0}(x,s)\) as \(\lambda\to\lambda_0\).

This theorem is easily obtained from Theorem 1, if one notes that \(v_\lambda(x,s)=u_\lambda(x,s)-u_{\lambda_0}(x,s)\) satisfies the equation \(\partial v/\partial s=L_\lambda v+\widetilde F(x,s,\lambda)\), where
\[ \widetilde F=(L_\lambda-L_{\lambda_0})u_{\lambda_0}+F(x,s,\lambda)-F(x,s,\lambda_0). \]

We note that Theorem 2 may be regarded as an extension of a theorem established by I. I. Gikhman \(\left({}^2\right)\) to equations of parabolic type. Just as Bogolyubov’s averaging principle \(\left({}^3\right)\) follows from Gikhman’s theorem, so from Theorem 2 there follows the following averaging principle for parabolic equations.

Let the function \(u_\varepsilon(x,s)\) satisfy the equation \(\partial u/\partial s=\varepsilon[L(x,s)u+F(x,s)]\) in the domain \(\Delta_\infty\), and let \(u_\varepsilon(x,0)=f(x)\).

Assume that \(L\) and \(F\) satisfy conditions \(1^0\) and \(2^0\) in the domain \(\Delta_\infty\). Suppose, moreover, that the following conditions are satisfied: 3) \(L(x,s)\) and \(F(x,s)\) are continuous in \(x\) uniformly with respect to \((x,s)\in\Delta_\infty\); 4) uniformly in \(x\in E_N\), the limits
\[ \overline L(x)=\lim_{t\to\infty}\frac{1}{t}\int_0^t L(x,s)\,ds,\qquad \overline F(x)=\lim_{t\to\infty}\frac{1}{t}\int_0^t F(a,s)\,ds; \]
exist; 5) there exists a solution of the Cauchy problem \(v(x,s)\) for the equation \(\partial v/\partial s=\overline L(x)v+\overline F(x)\), satisfying the condition \(v(x,0)=f(x)\).

Then for any \(T>0\) the relation
\[ \lim_{\varepsilon\to 0}\ \sup_{(x,s)\in\Delta_T}\left|u_\varepsilon(x,s/\varepsilon)-v(x,s)\right|=0 \]
holds.

Remark. If one makes the substitution \(s\varepsilon=t\), then the formulated principle may be regarded as a theorem on the limiting behavior of the solution of a parabolic equation all of whose coefficients oscillate rapidly. From this point of view the formulated result is close to \(\left({}^4\right)\).

1. Theorem 2 can also be applied to the study of the solution of an equation of the form
   \[ \frac{\partial u}{\partial s} = \varepsilon[L(x,s)u+F(x,s)] + \sum_{i=1}^{N} A_i(x)\frac{\partial u}{\partial x_i} \tag{3} \]
   by means of a suitable change of the unknown function.

In fact, introducing the function \(v_\varepsilon(x,t)=u_\varepsilon(Y(t,\varepsilon),t)\), we obtain for \(v_\varepsilon(x,t)\), generally speaking, an equation of the type \(\partial v_\varepsilon/\partial t=\varepsilon[L_1(x,t)v_\varepsilon+F_1(x,t)]\). (Here \(Y(t,x)=\{Y_1(t,x),\ldots,Y_N(t,x)\}\) is the solution of the system \(dY_i/dt=-A_i(Y_1,\ldots,Y_n),\ Y(0)=x\ (i=1,\ldots,N)\), and \(L_1\) is a certain elliptic operator of second order.)

This device makes it possible, in particular, to answer the question posed by A. N. Kolmogorov concerning the limiting transition from the invariant measure of a Markov process to the invariant measure of a dynamical system on the torus. The following theorem was stated by A. N. Kolmogorov as a hypothesis.

Theorem 3. Let \(X^{(\varepsilon)}\) be a diffusion process on the two-dimensional torus \(K\), whose transition-probability density is the Green’s function of the equation

\[ \frac{\partial u_\varepsilon}{\partial t} = \frac{1}{F(x)} \left( \frac{\partial u_\varepsilon}{\partial x_1} + \gamma\frac{\partial u_\varepsilon}{\partial x_2} \right) +\varepsilon L(x)u_\varepsilon \tag{4} \]

\[ \left( L(x)=\sum_{i,j=1}^{2} a_{ij}(x)\frac{\partial^2}{\partial x_i\partial x_j} +\sum b_i(x)\frac{\partial}{\partial x_i}; \quad F(x)>0. \right) \]

Let \(p_\varepsilon(x_1,x_2)\) be the density of the invariant measure of the process \(X^{(\varepsilon)}\). Then:

A. If \(\gamma\) is irrational, then, uniformly in \(x\in K\),

\[ p_\varepsilon(x)\to F(x)\Big/\iint_K F(x)\,dx_1\,dx_2 \quad \text{as } \varepsilon\to0. \]

B. If \(\gamma\) can be represented as an irreducible fraction \(m/n\), then, uniformly in \(x\in K\),

\[ p_\varepsilon(x)\to p_0(x)\,F(x)\Big/\int_0^1 F(x_1+tn,\ x_2+tm)\,dt \quad \text{as } \varepsilon\to0. \]

Here \(p_0(x)\) is the solution of the equation

\[ \sum_{i,j=1}^{2}\frac{\partial^2}{\partial y_i\partial x_j}(\widetilde a_{ij}p) - \sum_{i=1}^{2}\frac{\partial}{\partial x_i}(\widetilde b_i p) = 0, \tag{5} \]

satisfying the normalization condition,

\[ \widetilde a_{ij} = \int_0^1 a_{ij}(x_1+tn,\ x_2+tm)\,dt, \qquad \widetilde b_i = \int_0^1 b_i(x_1+tn,\ x_2+tm)\,dt. \]

We note that the function \(p_0\) so defined depends only on the difference \(x_2-\gamma x_1\), and therefore, for its actual computation, it is necessary to solve an ordinary differential equation of second order.

1. The method set forth above can also be applied to the study of elliptic differential equations with a small parameter multiplying the highest derivatives in the case where the trajectory of the limiting dynamical system does not reach the boundary of the domain. The results obtained in this way naturally complement the results that can be obtained by means of the method of (5).

Let, for example, the domain \(G_0=\{(x_1,x_2,x_3): r_1<x_3<r_2\}\) be a strip in the space \(E_3\), and let \(u_\varepsilon(x)\) be a solution of the elliptic equation \(\partial u_\varepsilon/\partial x_1+\gamma\partial u_\varepsilon/\partial x_2+\varepsilon(L(x)u_\varepsilon+F(x))=0\) in \(G_0\), satisfying the conditions \(u_\varepsilon(x_1,x_2,r_j)=c_j\ (j=1,2)\) (\(c_1\) and \(c_2\) are constants, the number \(\gamma\) is irrational). Suppose that all coefficients of the operator \(L\) and \(F\) are periodic in \(x_1\) and \(x_2\) with period 1, and that \(c(x)\leq 0\).

It is easy to verify that the function \(v_\varepsilon(x,t)=u_\varepsilon(x_1-t/\varepsilon, x_2-\gamma t/\varepsilon, x_3)\) satisfies a parabolic equation to which the method set forth above can be applied. Then we obtain \(u_\varepsilon(x)\to v_0(x_3)\) as \(\varepsilon\to 0\), uniformly in \(x\), where \(v_0(x_3)\) is the solution of the equation \(\overline{L}(x_3)v_0+\overline{F}(x_3)=0\) on the interval \(r_1<x_3<r_2\), and

\[ \overline{L}(x_3)=\int_0^1\int_0^1 L(x)\,dx_1dx_2,\qquad \overline{F}(x_3)=\int_0^1\int_0^1 F(x)\,dx_1dx_2. \]

Applying (2), one can also estimate the difference \(u_\varepsilon-u_0\). This estimate depends essentially on the arithmetic properties of the number \(\gamma\), and for no \(\gamma\) can it be made better than \(|u_\varepsilon-u_0|<K\varepsilon^\alpha\), where \(\alpha\) \((0<\alpha<1)\) is sufficiently small.

Applying to this same problem the method of \(({}^5,{}^6)\), we obtain the exact estimate \(|u_\varepsilon-u_0|<K\varepsilon\), if all the coefficients are sufficiently smooth and if, for some constant \(C(\gamma)\), for all integers \(m\) and \(n\) the inequality \(|n\gamma-m|\ge C(\gamma)/n^2\) holds (as is known \(({}^7)\), this relation holds for almost all irrational numbers \(\gamma\)). For irrational numbers that are “abnormally well” approximated by rational ones, the method of \(({}^5)\) is not applicable. In this case the estimate \(u_\varepsilon-u_0=O(\varepsilon)\), apparently, is also not true.

The author expresses gratitude to A. N. Kolmogorov for his attention to the work.

Moscow Forestry Engineering
Institute

Received
23 XI 1961

CITED LITERATURE

\({}^1\) Yu. V. Prokhorov, Theory of Probability and Its Applications, 1, No. 2, 177 (1956).
\({}^2\) I. I. Gikhman, Ukrainian Mathematical Journal, 4, No. 2, 215 (1952).
\({}^3\) N. N. Bogolyubov, On Certain Statistical Methods in Mathematical Physics, 1945.
\({}^4\) M. I. Vishik, L. A. Lyusternik, Uspekhi Matematicheskikh Nauk, 15, No. 4, 27 (1960).
\({}^5\) R. Z. Khasminskii, Doklady Akademii Nauk SSSR, 142, No. 3 (1962).
\({}^6\) A. N. Kolmogorov, Doklady Akademii Nauk SSSR, 93, No. 5, 753 (1953).
\({}^7\) A. Ya. Khinchin, Continued Fractions, Moscow, 1961.