Reports of the Academy of Sciences of the USSR

1. Volume 159, No. 2

MATHEMATICS

E. F. MISHCHENKO

ON THE PROBABILITY THAT A RANDOM POINT FALLS INTO A SMALL NEIGHBORHOOD OF A MOVING MANIFOLD

(Presented by Academician L. S. Pontryagin on 16 V 1964)

Let two objects move in the \(n\)-dimensional Euclidean space \(R^n\): a \(k\)-dimensional twice differentiable submanifold \(M\), changing its form and position according to the law

\[ M=M_s, \tag{1} \]

and a random point of Markov type, whose probability density \(p(\sigma,x,s,y)\) satisfies the Kolmogorov equation \((^1)\)

\[ \frac{\partial p}{\partial \sigma} + a^{ij}(\sigma,x)\frac{\partial^2 p}{\partial x^i\partial x^j} + b^i(\sigma,x)\frac{\partial p}{\partial x^i} =0. \tag{2} \]

Let its \(n\)-dimensional \(\varepsilon\)-neighborhood \(U(M)\) move together with \(M\). It is required to compute the probability that the random point enters the neighborhood \(U(M)\) during the time interval \(\sigma \leq s \leq \tau\).

In the present note the principal term of this probability is found. In the case when the manifold \(M\) is simply a controlled point, and \(U(M)\) is an \(n\)-dimensional ball of radius \(\varepsilon\) with center at this point, the problem was solved in \((^2,^3)\).

In what follows it is assumed that \(n-k \geq 3\).

In order to write down the formula obtained, we first make several constructions.

It is known (cf. \((^2)\)) that the sought probability \(\varphi(\sigma,x,\tau)\) (where \(x\) is the initial position of the random point at the time \(s=\sigma\)) is a solution of equation (2) under the conditions

\[ \varphi(\tau,x,\tau)=0, \]

\[ \varphi(\sigma,x,\tau)=1,\qquad x\in V(M_\sigma), \tag{3} \]

where \(V(M)\) is the boundary of the neighborhood \(U(M)\).

Through each point \(m_s\) of the manifold \(M_s\) draw the tangent plane \(P(m_s)\). Then choose \(n\) linearly independent vectors \(e_1,e_2,\ldots,e_n\), issuing from the point \(m_s\), so that: a) \(e_1,e_2,\ldots,e_k\) belong to \(P(m_s)\); b) in the coordinate system \(\xi^1,\xi^2,\ldots,\xi^n\), referred to the basis \(e_1,e_2,\ldots,e_n\), the differential operator

\[ a^{ij}(s,m_s)\frac{\partial^2}{\partial x^i\partial x^j} \tag{4} \]

is written in the form of the Laplace operator

\[ \sum_{\nu=1}^{n}\frac{\partial^2}{(\partial \xi^\nu)^2}. \tag{5} \]

The subspace conjugate to \(P(m_s)\), spanned by the vectors \(e_{k+1},\ldots,e_n\), will be denoted by \(Q(m_s)\).

The set of points of the subspace \(Q(m_s)\) at a distance \(\varepsilon\) (in the metric \(R^n\)) from the plane \(P(m_s)\) is an ellipsoid \(E_{m_s}\). Let its equation in the coordinates \(\xi\) be

\[ \sum_{i,j=k+1}^{n} c_{ij}\xi^i\xi^j=\varepsilon^2. \tag{6} \]

Obviously, up to small terms of higher order in \(\varepsilon\), we have
\[ V(M_s)=E_{m_s}\times M_s . \tag{7} \]

In what follows, denote by \(w(\xi^{k+1},\ldots,\xi^n)\) the harmonic function that vanishes as \(|\xi|\to\infty\) and is equal to unity on the ellipsoid \(E'_{m_s}\), singled out in \(Q(m_s)\) by the equation
\[ \sum_{i,j=k+1}^{n} c_{ij}\xi^i\xi^j=1 . \]

It is known that \(w\) can be represented in the form
\[ w=\frac{\alpha(m_s)}{\rho^{\,n-k-2}}+\Pi(\xi^{k+1},\ldots,\xi^n), \tag{8} \]
where
\[ \rho^2=(\xi^{k+1})^2+\cdots+(\xi^n)^2, \]
\(\alpha(m_s)\) is uniquely determined by the dimensions of the ellipsoid \(E'_{m_s}\), and \(\Pi\) is the double-layer potential produced by the ellipsoid \(E'_{m_s}\) at the point \((\xi^{k+1},\ldots,\xi^n)\). Differentiating the right- and left-hand sides of relation (8) in the direction \(\rho\) and then taking the integral over the surface \(E'_{m_s}\), we readily verify that
\[ \int_{E'_{m_s}} \frac{\partial w}{\partial \rho}\,dE'_{m_s} = \frac{4\pi^{(n-k)/2}}{\Gamma[(n-k)/2-1]}\,\alpha(m_s) = \beta(m_s), \tag{9} \]
where \(\Gamma\) is Euler’s gamma function.

We can now formulate the following proposition:

The solution of equation (2) under conditions (3) can be represented in the form
\[ \varphi(\sigma,x,\tau) = \varepsilon^{\,n-k-2} \int_{\sigma}^{\tau} ds \int_{M_s} p(\sigma,x,s,m_s)\,\beta(m_s)\,dM_s + \omega(\sigma,x,\tau,\varepsilon), \tag{10} \]
where \(\omega\) has magnitude of order \(\varepsilon^{\,n-k-1}\) for any point \(x\) separated from the manifold \(M_\sigma\) by a finite distance independent of \(\varepsilon\).

In formula (10) the inner integration is carried out over the entire manifold \(M_s\), the volume element in which is induced at each point by the frame \(e_1,e_2,\ldots,e_k\). It is easy to see that this definition of volume depends only on the coefficients \(a^{ij}\) of equation (2) and does not depend on the permissible arbitrariness in the choice of the frame \(e_1,\ldots,e_k\).

The scheme of proof of the proposition just formulated is as follows. The function
\[ \Phi(\sigma,x,\tau) = \varepsilon^{\,n-k-2} \int_{\sigma}^{\tau} ds \int_{M_s} p(\sigma,x,s,m_s)\,\beta(m_s)\,dM_s \tag{11} \]
is a solution of equation (2) outside the manifold \(M_\sigma\) and satisfies the first of conditions (3). But it does not satisfy the second, boundary condition (3). It turns out, however, that one can construct an \(n\)-dimensional ellipsoidal neighborhood of the manifold \(M_\sigma\) on whose boundary the values of the solutions \(\Phi(\sigma,x,\tau)\) and \(\varphi(\sigma,x,\tau)\) essentially coincide. Let us construct this neighborhood.

For this, in each subspace \(Q(m_\sigma)\) take the ellipsoid \(E^*_{m_\sigma}\) singled out by the equation
\[ \rho=\varepsilon, \tag{12} \]
and set
\[ V^*(M_\sigma)=E^*_{m_\sigma}\times M_\sigma . \tag{13} \]

The surface \(V^*\) is precisely the boundary of the neighborhood of the manifold \(M_\sigma\) that we need. We emphasize that, generally speaking, \(V^*\) does not coincide with \(V\).

Now, using the known asymptotic representations of the function \(p(\sigma, x, s, y)\) (cf., for example, \((^2)\)) and carrying out elementary, although rather cumbersome, calculations, we find that for \(x_0 \in V^*(M_\sigma)\)

\[ \Phi(\sigma, x_0, \tau)=\alpha(m_{0\sigma})+\omega_1(\sigma, x_0, \tau, \varepsilon), \tag{14} \]

where \(\omega_1\) is of order \(O(1)\) for \(\tau-\sigma \leqslant \varepsilon\) and vanishes as \(\varepsilon \to 0\) when \(\tau-\sigma>\varepsilon\). Here \(m_{0\sigma}\) denotes the projection of the point \(x_0\) onto the manifold \(M_\sigma\) in the direction of the plane \(Q(m_\sigma)\).

On the other hand, using a method which is a natural analogue of the method of \((^2)\), we can write the solution \(\varphi(\sigma, x, \tau)\) in a certain special form, from which it is directly seen that

\[ \varphi(\sigma, x_0, \tau)=\alpha(m_{0\sigma})+\omega_2(\sigma, x_0, \tau, \varepsilon), \tag{15} \]

where \(\omega_2\) has the same asymptotic character with respect to \(\varepsilon\) as \(\omega_1\).

Comparing relations (14) and (15), it is now not difficult to derive formula (10).

In conclusion we note that in the case \(n-k=2\) the simpler formula is valid

\[ \varphi(\sigma, x, \tau)=\frac{2\pi}{|\ln \varepsilon|} \int_{\sigma}^{\tau} ds \int_{M_s} p(\sigma, x, s, m_s)\,dM_s +o\!\left(\frac{1}{|\ln \varepsilon|}\right). \]

It is easily obtained if one uses a result of S. M. Nikol’skii \((^4)\).

Mathematical Institute named after V. A. Steklov
Academy of Sciences of the USSR

Received
9 V 1964

CITED LITERATURE

\(^1\) A. N. Kolmogoroff, Math. Ann., 104, 415 (1931).
\(^2\) E. F. Mishchenko, L. S. Pontryagin, Izv. AN SSSR, ser. matem., 25, 477 (1961).
\(^3\) A. N. Kolmogorov, E. F. Mishchenko, L. S. Pontryagin, DAN, 145, No. 5 (1962).
\(^4\) S. M. Nikol’skii, Theory of Probability and Its Applications, 9, issue 2 (1964).