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UDC 519.27 + 517.91
MATHEMATICS
R. Z. KHAS’MINSKII
SOME LIMIT THEOREMS FOR SOLUTIONS OF DIFFERENTIAL EQUATIONS WITH A RANDOM RIGHT-HAND SIDE
(Presented by Academician A. N. Kolmogorov on 14 IX 1965)
In many physical and, in particular, radio-engineering problems it is of interest to study random processes that are solutions of differential equations whose right-hand side contains randomness. In this case it is often possible to single out in the right-hand side of the equation a small parameter \(\varepsilon\), which may characterize the smallness of the random perturbation, its “correlation time,” etc. (see \((^1)\)). In the present note we study the asymptotic behavior of such random processes as \(\varepsilon \to 0\).
- Let \(F(x,t,\omega,\varepsilon)\) be a function with values in \(l\)-dimensional Euclidean space \(E^l\), defined for \(x \in E^l\), \(t \ge 0\), \(\omega \in \Omega\), \(\varepsilon \ge 0\); here \(\Omega\) is the space of elementary events, on the \(\sigma\)-algebra \(\mathfrak A\) of measurable subsets of which a probability measure \(P\) is given. Suppose that \(F(x,t,\omega,\varepsilon)\), for fixed \(x,\varepsilon\), is a random process measurable in \(t,\omega\), satisfies the Lipschitz condition
\[ |F(x_2,t,\omega,\varepsilon)-F(x_1,t,\omega,\varepsilon)|<L|x_2-x_1| \tag{1} \]
and, for all \(t>0\), the condition
\[ \mathbf P\left\{\int_0^t |F(0,s,\omega)|\,ds<\infty\right\}=1. \tag{2} \]
When these requirements are fulfilled, the problem
\[ dx/dt=\varepsilon F(x,t,\omega,\varepsilon);\qquad x(0)=x_0, \tag{3} \]
has a unique solution, a random process \(x_\varepsilon(t,\omega)\) that is continuous with probability 1.
The asymptotic behavior, as \(\varepsilon \to 0\), of the solution \(x_\varepsilon(t,\omega)\) of problem (3) has been considered, for example, in \((^{1-3})\).
In \((^3)\) it is proved that the function \(x_\varepsilon(t,\omega)\) on a time interval of order \(O(1/\varepsilon)\) can be uniformly approximated by the solution of the problem
\[ \frac{dx}{dt}=\varepsilon\Phi^{(0)}(x);\qquad x(0)=x_0 \quad \left(\Phi^{(0)}(x)=\lim_{T\to\infty}\frac1T\int_0^T M F^{(0)}(x,t,\omega)\,dt\right), \]
if, uniformly in \(x,t,\omega\) as \(\varepsilon \to 0\), the relation \(F(x,t,\omega,\varepsilon)=F^{(0)}(x,t,\omega)+o(1)\) holds, and the process \(F^{(0)}(x,t,\omega)\) satisfies the law of large numbers.
It follows from this result that, when \(\Phi^{(0)}(x)\equiv 0\), the random process \(x_\varepsilon(\varepsilon t,\omega)\) converges in probability to zero as \(\varepsilon \to 0\). R. L. Stratonovich in \((^1)\) first drew attention to the fact that in the still “slower” time \(\tau=\varepsilon^2 t\) there is convergence to a Markov process. However, some points in the proof of this assertion given in \((^2)\), when \(F\) is a stationary-in-time random process, seem to us unconvincing. We shall consider this problem under other assumptions and by another method.
Let the function \(F(x,t,\omega,\varepsilon)\) satisfy the conditions:
B\(_1\). Uniformly in \(x,t,\omega\), except perhaps for a set of \(\omega\)-values of probability 0, the relations
\[ F(x,t,\omega,\varepsilon) = F^{(0)}(x,t,\omega)+\varepsilon F^{(1)}(x,t,\omega)+o(\varepsilon) \quad (\varepsilon \to 0), \]
\[ |F^{(i)}|,\ |\partial F^{(i)}/dx|,\ |\partial^2 F^{(i)}/\partial x_j\partial x_k|<C \quad (i=0,1;\ j,k=1,\ldots,l). \]
B\(_2\). There exists a family of \(\sigma\)-algebras \(N_s^t\) \((0\le s\le t\le \infty)\) of subsets of \(\Omega\) such that \(N_s^t\subset \mathfrak A\); \(N_s^t\subset N_{s_1}^{t_1}\) if \(s_1\le s,\ t\le t_1\), and for all \(x\in E_n,\ t\ge 0\) the random variables \(F^{(i)}(x,t,\omega)\) are \(N_t^t\)-measurable. (For example, if \(F^{(i)}(x,t,\omega)=F^{(i)}(x,\xi(t,\omega))\), then one may take as \(N_s^t\) the \(\sigma\)-algebra of events generated by events of the form \(\{\xi(u,\omega)\in A\}\), \(s\le u\le t\).) Moreover, for any \(t\ge 0\), \(B\in N_{t+\tau}^{\infty}\), and for some function \(\beta(\tau)\) such that, as \(\tau\to\infty\), the function \(\tau^6\beta(\tau)\downarrow 0\), with probability 1,
\[ \left|P\{B/N_0^t\}-P(B)\right|<\beta(\tau). \tag{4} \]
(Condition (4) was considered by Ibragimov in \((^{4,5})\).)
B\(_3\). Uniformly in \(x,t_0>0\) the limits
\[ \lim_{T\to\infty}\frac1T\int_{t_0}^{t_0+T} MF^{(1)}(x,t,\omega)\,dt = \Phi^{(1)}(x), \]
\[ \lim_{T\to\infty}\frac1T \int_{t_0}^{t_0+T}\int_{t_0}^{t_0+T} \operatorname{cov}\bigl(F_j^{(0)}(x,s,\omega),F_k^{(0)}(x,t,\omega)\bigr)\,ds\,dt = a_{jk}(x), \tag{5} \]
\[ \lim_{T\to\infty}\frac1T \int_{t_0}^{t_0+T} ds \int_{t_0-T}^{s} M\left\{ \frac{\partial F^{(0)}}{\partial x}(x,s,\omega)F^{(0)}(x,t,\omega) \right\}\,dt = K(x), \]
exist, and the integrals
\[ \int_0^T MF^{(0)}(x,t,\omega)\,dt \quad\text{and}\quad \int_0^T \frac{\partial}{\partial x}MF^{(0)}(x,t,\omega)\,dt \]
are bounded uniformly in \(x,T\).
B\(_4\). There exists a sequence \(T_n\to\infty\), growing no faster than a geometric progression and such that, as \(n\to\infty\),
\[ \delta(T_n)=\sup_{x\in E^l,\ t_0>T_n} \left| T_n^6 \int_{t_0}^{t_0+T_n} MF^{(0)}(x,t,\omega)\,dt \right| \downarrow 0. \]
Theorem 1. If the function \(F(x,t,\omega,\varepsilon)\) satisfies conditions B\(_1\)—B\(_2\), then the process \(x_\varepsilon(\varepsilon^2 t)\) on the time interval \(0\le \varepsilon^2 t\le \tau_0\) converges weakly as \(\varepsilon\to 0\) to a continuous, with probability 1, Markov process \(X^{(0)}(t,\omega)\), for which
\[ M\{\Delta X^{(0)}(t,\omega)\mid X^{(0)}(t,\omega)=x\} = \bigl(K(x)+\Phi^{(1)}(x)\bigr)\Delta t+o(\Delta t), \]
\[ M\{\Delta X_j^{(0)}(t,\omega)\Delta X_k^{(0)}(t,\omega)\mid X^{(0)}(t,\omega)=x\} = a_{jk}(x)\Delta t+o(\Delta t). \]
From Theorem 1 one can easily derive
Theorem 2. Let the function \(F\) satisfy conditions B\(_1\), B\(_2\), and let
\[
F^{(i)}(x,t,\omega)=F^{(i)}(x,\xi(t,\omega)),
\]
where \(\xi(t,\omega)\) is a random process periodic with period \(\theta\). Suppose also that the condition
\[ \int_0^\theta MF^{(0)}(x,t,\omega)\,dt=0 \tag{6} \]
is satisfied.
Then the conclusion of Theorem 1 is valid, where \(a_{jk}(x)\), \(\Phi^{(1)}(x)\), and \(K(x)\) are computed by the formulas
\[ \Phi^{(1)}(x)=\frac1\theta\int_0^\theta MF^{(1)}(x,t,\omega)\,dt; \]
\[ a_{jk}(x)=\frac{1}{\theta}\int_0^\theta ds\int_{-\infty}^{+\infty} \operatorname{cov}\left(F_j^{(0)}(x,s,\omega),F_k^{(0)}(x,t,\omega)\right)dt, \]
\[ K(x)=\frac{1}{\theta}\int_0^\theta ds\left[ \int_{-\infty}^{\theta} \operatorname{cov}\left(\frac{\partial F^{(0)}}{\partial x}(x,s,\omega), F^{(0)}(x,s+u,\omega)\right)du + \int_0^s \frac{\partial M F^{(0)}}{\partial x}(x,s)M F^{(0)}(x,t)\,dt \right]. \tag{7} \]
Remark 1. For a stationary process \(F\), formula (7) is somewhat simplified (since condition (6) becomes the condition \(MF^{(0)}(x,t,\omega)\equiv 0\), and in (7) one can pass to the limit as \(\theta\to 0\)). For this case they were obtained by Stratonovich in \((^1,^2)\).
Remark 2. Relying on Theorem 1, one can show that the conclusion of Theorem 2 is also valid in the case when the process \(\xi(t,\omega)\) is not periodic, but only converges to a periodic one in a sufficiently weak sense as \(t\to\infty\).
Remark 3. Theorem 1 can be regarded as a generalization of the central limit theorem for random processes satisfying a mixing condition in one form or another (cf. \((^4,^5)\)). This becomes clear if one sets \(F(x,t,\omega)\equiv F(t,\omega)\).
- Theorems 1 and 2 can be used, in particular, for a rigorous justification, refinement, and indication of the conditions of applicability of certain conclusions of \((^1)\). Let us consider, for example, the question of parametric excitation of linear systems by random forces (see \((^1)\), § 19).
The equation of motion of a system whose frequency undergoes small random perturbations \(\varepsilon\xi(t,\omega)\), and whose friction coefficient is \(\gamma\), has the form
\[ \ddot{x}+\mu^2(1+\varepsilon\xi(t,\omega))x+\gamma\dot{x}=0. \tag{8} \]
If the process \(\xi(t,\omega)\) is stationary, ergodic, and has zero mean, with \(|\xi(t,\omega)|<C\) with probability 1, and \(\gamma=\varepsilon\gamma_1\) (\(\gamma_1=\mathrm{const}\)), then, making the change of variables
\[ x=e^u\cos(\mu t+\theta),\qquad \dot{x}=-\mu e^u\sin(\mu t+\theta) \tag{9} \]
and applying Theorem 1.1 from \((^3)\), we obtain that system (8) is stable on a time interval \(O(1/\varepsilon)\) for all \(\gamma_1>0\), and the approximation to the equilibrium can be described by the equation \(\ddot{x}+\mu^2x+\varepsilon\gamma_1\dot{x}=0\) the more accurately, the smaller \(\varepsilon\) is.
A more interesting case is obtained if \(\gamma=\varepsilon^2\gamma_1\), where \(\gamma_1=\mathrm{const}\). Then the stability or instability of the system can be detected only on a time interval \(O(1/\varepsilon^2)\). If, in addition to the requirements listed above, the process \(\xi(t,\omega)\) satisfies condition B2, then Theorem 2 can be applied to the system of equations for the process \(\{u_\varepsilon(t,\omega),\theta_\varepsilon(t,\omega)\}\) obtained after the substitution (9) from (8). This theorem makes it possible to establish that on the interval \(0\leq \varepsilon^2t\leq \tau_0\), as \(\varepsilon\to 0\),
\[ \{u_\varepsilon(t,\omega),\theta_\varepsilon(t,\omega)\}\to \{u_0(\varepsilon^2t,\omega),\theta_0(\varepsilon^2t,\omega)\}, \]
where \(u_0(\tau,\omega)\) and \(\theta_0(\tau,\omega)\) are mutually independent Markov processes of diffusion type with constant diffusion and drift coefficients. In particular, the process \(u_0(\tau,\omega)\) has diffusion coefficient \(\mu^2 f(2\mu)/16\) and drift coefficient \(\mu^2 f(2\mu)-4\gamma_1/8\) (here \(f(\lambda)\) is the spectral density of the process \(\xi(t,\omega)\)). Hence it is clear that, if
\[ \mu^2 f(2\mu)/4\geq \gamma_1, \tag{10} \]
system (8) is unstable for sufficiently small \(\varepsilon\). If the opposite condition is fulfilled, Theorem 2 permits one to assert only that no loss of stability will occur on a time interval of order \(O(1/\varepsilon^2)\). Condition (10) was obtained in \((^1)\) by means of nonrigorous considerations.
- An important point in the proof of Theorem 1 is the establishment of the estimate
\[ M\left|x_\varepsilon(\tau+h,\omega)-x_\varepsilon(\tau,\omega)\right|^4<ch^2, \]
which guarantees compactness of the family of distributions associated with the process \(x_\varepsilon(\tau,\omega)\) in the space of continuous functions (see (6)). For the establishment of this estimate the following is essential.
Lemma 1. Let \(\{\xi_n^{(1)},\ldots,\xi_n^{(2k)}\}\) \((n=1,2,\ldots)\) be a sequence of random vectors in \(E^{(2k)}\) with zero mathematical expectation, satisfying Rosenblatt’s strong mixing condition with mixing coefficient \(\alpha(\tau)\). Suppose also that either the conditions
\[ \left|\xi_n^{(i)}\right|<C,\qquad \int_0^\infty \tau^{k-1}\alpha(\tau)\,d\tau<\infty, \]
or, for some \(m>2\), the conditions
\[ M\left|\xi_n^{(i)}\right|^{m(2k-1)}<C,\qquad \int_0^\infty \tau^{k-1}[\alpha(\tau)]^{(m-2)/m}\,d\tau<\infty \]
are fulfilled. Then for all \(N>0\) the estimate
\[ \sum_{n_i=1}^{N}\left|M\left(\xi_{n_1}^{(1)}\cdots \xi_{n_{2k}}^{(2k)}\right)\right|<C_1N^k . \tag{11} \]
holds.
The proof of this lemma is close in idea to the proof of similar assertions in (7). We note that in the case \(\xi_n^{(1)}=\xi_n^{(2)}=\cdots=\xi_n^{(2k)}=\xi_n\), from (11) there follows the estimate
\[ M(\xi_1+\ldots+\xi_N)^{2k}<C_1N^k, \]
which, as is known, cannot be improved even for independent random variables.
For estimating the conditional moments of first and second order of the increment of the process \(x_\varepsilon(\varepsilon^2t)\), the following lemma, easily following from (4), is useful.
Lemma 2. Let the family of \(\sigma\)-algebras \(N_s^t\) satisfy condition (4), let \(\xi\) be an \(N_0^t\)-measurable random variable, and let \(G(x,\omega)\), \(|G|<1\), be a measurable function on \(E^l\times\Omega\) which, for each fixed \(x\), is an \(N_{t+\tau}^{\infty}\)-measurable random variable. Let \(g(x)=MG(x,\omega)\). Then with probability 1
\[ \left|M\{G(\xi(\omega),\omega)/N_0^t\}-g(\xi(\omega))\right|<2\beta(\tau). \]
Institute for Problems of Information Transmission
Academy of Sciences of the USSR
Received
10 IX 1965
CITED LITERATURE
- R. L. Stratonovich, Selected Problems in the Theory of Fluctuations in Radio Engineering, Moscow, 1961.
- R. L. Stratonovich, Conditional Markov Processes, Moscow, 1966.
- R. Z. Khas’minskii, Theory of Probability and Its Applications, 11, 2 (1966).
- I. A. Ibragimov, DAN, 125, No. 4, 711 (1959).
- I. A. Ibragimov, Theory of Probability and Its Applications, 7, 4, 316 (1962).
- Yu. V. Prokhorov, Theory of Probability and Its Applications, 1, 2, 176 (1956).
- S. N. Bernstein, Uspekhi Mat. Nauk, vol. 10, 65 (1944).