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Reports of the Academy of Sciences of the USSR
- Volume 145, No. 5
MATHEMATICS
A. M. Ilyin and R. Z. Khasminskii
ON THE ERGODIC PROPERTY OF NONHOMOGENEOUS DIFFUSION PROCESSES
(Presented by Academician I. G. Petrovsky, 22 III 1962)
Let the operator
\[ L(t,x)=\sum_{i,j=1}^{N} a_{ij}(t,x)\frac{\partial^2}{\partial x_i\partial x_j} +\sum_{i=1}^{N} b_i(t,x)\frac{\partial}{\partial x_i} \]
be defined everywhere in the domain \(\{x\in E_N\}\times\{-\infty<t<\infty\}\) (\(E_N\) is \(N\)-dimensional Euclidean space). Suppose that the coefficients satisfy the conditions:
\[ \sum_{i,j=1}^{N} a_{ij}(t,x)\xi_i\xi_j \ge \gamma(x)\sum_{i=1}^{N}\xi_i^2 \]
for all real \(\xi_i\) and for some continuous positive function \(\gamma(x)\);
\[ |a_{ij}(t,x)|\le M(r^2+1), \qquad |b_i(t,x)|<M(r+1) \]
for all \(t\), where \(M\) is a constant, and
\[ r^2=\sum_{i=1}^{N}x_i^2=|x|^2. \]
The coefficients are also assumed to be sufficiently smooth.
As is known, with the operator \(L(t,x)\) there is associated a Markov process \(X(t,\omega)\) in \(E_N\), the transition probability density \(p(s,x,t,y)\) of which is the Green’s function of the equations:
\[ \frac{\partial u}{\partial s}+L(s,x)u=0; \tag{1} \]
\[ \frac{\partial u}{\partial t}=L^*(t,y)u. \tag{2} \]
We shall show that under the condition
\[ \sum_{i=1}^{N}\left[a_{ii}(t,x)+b_i(t,x)x_i\right]<-\delta<0 \quad \text{for } r>r_1 \tag{3} \]
the probability distribution “does not spread out” as time passes. Under additional restrictions on the coefficients it turns out that
\[ p(s,x,t,y)-p(s,z,t,y)\to 0 \]
as \(t-s\to\infty\), for arbitrary \(x\) and \(z\) from \(E_N\).
Theorem 1. If the coefficients satisfy condition (3), then there exists
\[ \lim_{s\to-\infty} p(s,x,t,y)=p(t,y)>0. \]
It is not difficult to show that the function \(p(t,y)\), like \(p(s,x,t,y)\), satisfies the Chapman–Kolmogorov relation and, consequently, equation (2) for all \(t\) \((-\infty<t<\infty)\). This function is obviously bounded if the following condition is fulfilled:
A. There exists a number \(M\) such that, for all \(s,x\), and \(y\), the inequality
\[ p(s,x,s+1,y)<M \]
holds.
However, even when this condition is fulfilled, equation (2) may have several bounded positive solutions, defined for all \(t\) and such that
\[ \int_{E_N} p(t,y)\,dy=1. \tag{4} \]
For example, the equation
\[ \frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial y^2}+\frac{\partial}{\partial y}(yu) \]
has the solutions
\[ u_1=\frac{1}{\sqrt{2\pi}}\exp\left(-\frac12 y^2\right) \quad\text{and}\quad u_2=\frac{1}{\sqrt{2\pi}}\exp\left\{-\frac12 (y-e^{-t})^2\right\}. \]
It is clear that, in order to single out the unique bounded everywhere solution of equation (2), this solution must be subject to an additional condition.
Theorem 2. Suppose that conditions \(A^*\) and (3) are satisfied. Then the function \(p(t,y)\) is the unique bounded solution of equation (2) in the domain \(-\infty<t<\infty\), for which (4) is satisfied and the integral in (4) converges uniformly.
In this case the process has the ergodic property (see (1)):
Theorem 3. Suppose that conditions \(A\) and (3) are satisfied, and let \(K\) be an arbitrary compact set in \(E_N\). Then
\[ p(s,x,t,y)-p(s,z,t,y)\to 0, \]
\[ \int_{E_N}\left|p(s,x,t,y)-p(s,z,t,y)\right|\,dy\to 0 \tag{5} \]
as \(t\to\infty\), uniformly with respect to \(x\) and \(z\) from \(K\). Relations (5) remain valid if the function \(p(s,z,t,y)\) is replaced by the function \(p(t,y)\).
It is clear that the limiting behavior of the probability distribution density \(p(s,x,t,y)\) is closely connected with the asymptotics, for large values of time, of the solutions of the Cauchy problem for equations (1), (2).
Theorem 4. Suppose that conditions \(A\) and (3) are satisfied, and let \(v(s,x)\) be the solution of the Cauchy problem for \(s\leq t\) for equation (1), satisfying the condition
\[ v(t,x)=v_0(x). \tag{6} \]
Then
\[ \lim_{s\to-\infty} v(s,x)=\int_{E_N} p(t,y)v_0(y)\,dy, \]
and the convergence is uniform in each compact set \(K\subset E_N\).
Theorem 5. Suppose that conditions \(A\) and (3) are satisfied; let \(u(t,y)\) be the solution of the Cauchy problem for \(t\geq s\) for equation (2) with the initial condition
\[ u(s,y)=u_0(y). \tag{7} \]
If
\[ \int_{E_N}|u_0(y)|\,dy<\infty, \]
then
\[ u(t,y)-p(t,y)\int_{E_N}u_0(y)\,dy\to 0 \quad\text{as } t\to\infty \]
uniformly in each compact set \(K\subset E_N\). If
\[ \int_{E_N}u_0(y)\,dy=\infty, \]
then
\[ u(t,y)\xrightarrow[t\to\infty]{}\infty. \]
The proof of Theorems 1 and 4, which are essentially equivalent, is carried out according to the following plan. Let \(B\) be some sphere in \(E_N\), and suppose that condition (3) is satisfied. Denote by \(C(t,T)\) the event consisting in the fact that the trajectory \(X(t,\omega)\) of the process under consideration visits the set \(B\) at least once on the time interval \([t,t+T]\). One can prove the following assertion, fundamental for the proof of Theorem 4 (an analogous lemma was proved by A. N. Kolmogorov in \((^2)\) for homogeneous Markov chains with a countable number of states):
Lemma. For any \(\varepsilon>0\) and compact set \(K\) there exist a number \(T\), depending only on \(\varepsilon\), and a number \(S\), depending only on \(K\), such that
\[ \mathbf P\{C(t,T)/X(s,\omega)=x\}>1-\varepsilon \]
for all \(x\in K\), \(t>S+s\).
* It is clear that condition \(A\) is satisfied, for example, if all the coefficients of equations (1) and (2) are bounded together with their first derivatives.
From this lemma two corollaries follow easily.
- Let \(G\) be an open set in \(E_N\). Then there exists \(\delta_G>0\) such that
\[ \mathbf P\{X(t,\omega)\in G/X(s,\omega)=x\}>\delta_G, \]
if \(x\in K,\ t-s>T(K)\).
- For every \(\varepsilon>0\) there exists a compact set \(K_1\) such that
\[ \mathbf P\{X(t,\omega)\in K_1/X(s,\omega)=x\}>1-\varepsilon, \]
if \(x\in K,\ t>s>T(K)\).
If \(v(s,x)\) is a solution of problem (1), (6), then one can choose a sequence \(s_k\to-\infty\) such that the sequence \(v(s_k,x)\) tends to \(\varphi(x)\) uniformly on every compact set. Relying on corollaries 1 and 2, it is not hard to show that \(\varphi(x)\equiv \mu=\mathrm{const}\) and \(\mu>0\), if \(v_0(x)\ge 0\) and \(v_0(x)\not\equiv 0\). Consequently, \(v(s,x)\to\mu\) as \(s\to-\infty\).
Studying further the solutions of problems (1), (6) and (2), (7) and using Green’s formula for equations (1), (2), one can determine how this limiting value \(\mu\) is connected with the solution of equation (2), and prove Theorems 2, 3, and 5.
Let us note that condition (3) is fairly sharp, since when the condition
\[ \sum_{i=1}^{N}(a_{ii}+b_i x_i)>\delta>0 \]
is satisfied for \(r>r_1\), the distribution density \(p(s,x,t,y)\to 0\) as \(s\to-\infty\) \((^3)\). If the process \(X(t,\omega)\) is considered on a compact manifold \(M\), then the conclusions of Theorems 1–5 are valid under minimal restrictions on the coefficients. It suffices, for example, to assume that all coefficients are continuous, bounded, and
\[ \sum_{i,j=1}^{N} a_{ij}\xi_i\xi_j>m\sum_{i=1}^{N}\xi_i^2 \]
for all real \(\xi_i\) and \((t,x)\in(-\infty,\infty)\times M\).
The results obtained also extend to the case of a bounded domain in \(E_N\), if on the boundary of this domain the solution of equation (1) satisfies the condition \(\partial u/\partial \nu=0\) (\(\nu\) is the conormal to the boundary of the domain), while the solution of equation (2) satisfies the adjoint boundary condition.
In the case of homogeneous Markov processes \((L=\tilde L(x))\), questions analogous to those considered in the present note were studied, under more general assumptions, in \((^4)\). Let us note in conclusion that a theorem similar to Theorem 4 was proved earlier for the one-dimensional case by Ya. I. Kanel \((^5)\).
Received
20 III 1962
REFERENCES
\(^1\) A. N. Kolmogorov, UMN, 5, 5 (1938).
\(^2\) A. N. Kolmogorov, Bull. Moscow Univ., 1, A, issue 3, 1 (1937).
\(^3\) A. M. Il’in, UMN, 16, 2, 115 (1961).
\(^4\) R. Z. Khas’minskii, Theory of Probability and Its Applications, 5, 2, 196 (1960).
\(^5\) Ya. I. Kanel, DAN, 136, No. 2, 277 (1961).