UDC 517.946.43

A CERTAIN PROBLEM WITH A SMALL PARAMETER FOR PARABOLIC DIFFERENTIAL EQUATIONS

E. F. MISHCHENKO, M. S. NIKOLSKII

In the present paper a complete exposition is given of the results published in the note [1].

§ 1. FORMULATION OF THE PROBLEM. BASIC ESTIMATES

Let two objects move in the \(n\)-dimensional Euclidean space \(R^n\): a \(k\)-dimensional twice continuously differentiable manifold \(M\), changing its shape and position according to the law \(M = M_s\), and a random point of Markov type, whose probability density \(p(\sigma, x, \tau, y)\) is subject to the Kolmogorov equation [2]

\[ \frac{\partial p}{\partial \sigma} + a^{ij}(\sigma, x)\frac{\partial^2 p}{\partial x^i \partial x^j} + b^i(\sigma, x)\frac{\partial p}{\partial x^i} = 0 . \tag{1} \]

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 \(U(M)\) in the time interval \(\sigma \leq s \leq \tau\).

In the present article 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 the works [3–5]*.

In the first three paragraphs of the present paper we assume that the dimensions of the space \(R^n\) and of the manifold \(M\) satisfy the relation \(n-k \geq 3\).

For the purpose of simplifying the notation and the computations, at first we shall carry out the discussion assuming the coefficients of equation (1) to be constant and the manifold \(M\) to be fixed.

It is known that the desired probability \(\varphi(\sigma, x, \tau)\) (here \(x\) is the initial position of the random point at the time \(s=\sigma\)) is the solution of equation (1) under the conditions

\[ \varphi(\tau, x, \tau) = 0, \qquad \varphi(\sigma, x, \tau) = 1 \quad \text{for } x \in V(M), \tag{2} \]

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

* In paper [4] in formula (3) and in paper [5] in formula (2), under the integral sign the factor \(\sqrt{a(s)}\), where \(a(s)=\det\|a^{ij}(s,z(s))\|\), was omitted.

Through each point \(m\in M\) draw the tangent plane \(P(m)\). Then choose \(n\) linearly independent vectors \(e_1,\ldots,e_n\), issuing from the point \(m\), so that: a) \(e_1,\ldots,e_k\) belong to \(P(m)\); b) in the coordinate system \(\xi^1,\ldots,\xi^n\) referred to the basis \(e_1,\ldots,e_n\), the differential operator
\(a^{ij}\dfrac{\partial^2}{\partial x^i\partial x^j}\) is written in the form of the Laplace operator \(\displaystyle \sum_{\nu=1}^{n}\dfrac{\partial^2}{(\partial \xi^\nu)^2}\).

The subspace conjugate to \(P(m)\), spanned by the vectors \(e_{k+1},\ldots,e_n\), will be denoted by \(Q(m)\). The set of points of the subspace \(Q(m)\) that are at distance \(\varepsilon\) in the metric \(R^n\) from the plane \(P(m)\) is an ellipsoid \(E_m\). Let its equation in the coordinates \(\xi\) be

\[ \sum_{i,j=k+1}^{n} c_{ij}(m)\xi^i\xi^j=\varepsilon^2 . \tag{3} \]

Obviously, up to small quantities of higher order in \(\varepsilon\), we have

\[ V(M)=E_m\times M . \tag{4} \]

Denote by \(w(\xi^{k+1},\ldots,\xi^n)\) the harmonic function tending to \(0\) as \(|\xi|\to\infty\) and equal to one on the ellipsoid \(E_m\). It is known that \(w\) can be represented in the form

\[ w=-\frac{\alpha(m)}{r^{\,n-k-2}_{(\xi)}}\,\varepsilon^{\,n-k-2} +\pi_m(\xi^{k+1},\ldots,\xi^n), \tag{5} \]

where \(r^2(\xi)=(\xi^{k+1})^2+\ldots+(\xi^n)^2\); \(\alpha(m)\) does not depend on \(\varepsilon\) (see [3]); \(\pi_m\) is the double-layer potential created by the ellipsoid \(E_m\).

In a sufficiently small neighborhood of \(M\), for each point \(x\) its projection \(m_x\) onto the manifold \(M\) in the direction of the plane \(Q(m_x)\) is uniquely determined. Thus, to a point \(x\) from this neighborhood there correspond uniquely the coordinates \(\xi_x^{k+1},\ldots,\xi_x^n\), and one may define the function

\[ \rho(x)=\sqrt{(\xi_x^{k+1})^2+\ldots+(\xi_x^n)^2}. \]

The subsequent considerations will mainly be carried out in this neighborhood, which we shall denote by \(R_0\), i.e. for \(\rho(x)\le r_0\), where \(r_0\) is some positive constant.

Define the function \(W^*(x)\):

\[ W^*(x)=-\frac{\alpha(m_x)}{\rho^{\,n-k-2}(x)}\,\varepsilon^{\,n-k-2} +\pi_{m_x}(\xi_x^{k+1},\ldots,\xi_x^n) = \]

\[ =-\frac{A^*(x)}{\rho^{\,n-k-2}(x)}\,\varepsilon^{\,n-k-2}+\Pi^*(x). \tag{6} \]

Let us note two inequalities:

\[ \left|-\frac{A^*(x)}{\rho^{\,n-k-2}(x)}\,\varepsilon^{\,n-k-2}\right| \le \frac{B_1\varepsilon^{\,n-k-2}}{\rho^{\,n-k-2}(x)}, \tag{7} \]

\[ |\Pi^*(x)|\le \frac{B_2\varepsilon^{\,n-k-1}}{\rho^{\,n-k-1}(x)}, \tag{8} \]

where \(B_1,B_2\) are constants. The first inequality is obvious. The second is proved in the same way as in [3].

In what follows, we shall agree to denote constants by the letter \(B\) with an index. Denote by \(L\) the differential operator

\[ L=-a^{ij}\frac{\partial^2}{\partial x^i\partial x^j}-b^i\frac{\partial}{\partial x^i}. \tag{9} \]

Using the fact that both terms in formula (6) are solutions of the Laplace equation in the subspace \(Q(m)\), and using the special choice of the subspace \(Q(m)\), one can prove the inequalities

\[ \left|L\left[\frac{A^*(x)}{\rho^{\,n-k-2}(x)}\varepsilon^{\,n-k-2}\right]\right| \leq \frac{B_3\varepsilon^{\,n-k-2}}{\rho^{\,n-k-1}(x)}, \tag{10} \]

\[ |L\Pi^*(x)|\leq \frac{B_4\varepsilon^{\,n-k-1}}{\rho^{\,n-k}(x)}. \tag{11} \]

Smoothly reduce the functions \(A^*(x)\), \(\Pi^*(x)\) to zero near the surface \(\rho(x)=r_0\), and extend them by zero for \(x\) lying outside \(R_0\). Denote the resulting functions respectively by \(A(x)\), \(\Pi(x)\). Thus, in the space \(R^n\) the function

\[ W(x)=\frac{A(x)}{\rho^{\,n-k-2}(x)}\varepsilon^{\,n-k-2}+\Pi(x) \tag{12} \]

is defined.

Obviously, the smoothing can be carried out so that, for \(x\in R_0\), estimates of the type of inequalities (7), (8), (10), (11) will be valid:

\[ \left|\frac{A(x)}{\rho^{\,n-k-2}(x)}\varepsilon^{\,n-k-2}\right| \leq \frac{B_5\varepsilon^{\,n-k-2}}{\rho^{\,n-k-2}(x)}, \tag{13} \]

\[ |\Pi(x)|\leq \frac{B_6\varepsilon^{\,n-k-1}}{\rho^{\,n-k-1}(x)}, \tag{14} \]

\[ \left|L\left[\frac{A(x)}{\rho^{\,n-k-2}(x)}\varepsilon^{\,n-k-2}\right]\right| \leq \frac{B_7\varepsilon^{\,n-k-2}}{\rho^{\,n-k-1}(x)}, \tag{15} \]

\[ |L\Pi(x)|\leq \frac{B_8\varepsilon^{\,n-k-1}}{\rho^{\,n-k}(x)}. \tag{16} \]

Denote by \(q_\varepsilon(\sigma,x,\tau,y)\) the Green function of the exterior boundary-value problem for equation (1) and the surface \(V(M)\), defined by equality (4). Using the known formula for the solution of an inhomogeneous parabolic equation, we can write the required probability in the following form:

\[ \varphi(\sigma,x,\tau)=W(x)-\int_{R_\varepsilon}q_\varepsilon(\sigma,x,\tau,y)\cdot W(y)\,dy- \]

\[ -\int_\sigma^\tau ds\int_{R_\varepsilon}q_\varepsilon(\sigma,x,s,y)\cdot L[W(y)]\,dy, \tag{17} \]

where \(R_\varepsilon=R_0-U(M)\).

It is known that the inequality

\[ q_\varepsilon(\sigma,x,\tau,y)\leq p(\sigma,x,\tau,y) \tag{18} \]

holds.

Consider the functions

\[ \omega_l(\sigma,x,\tau)=\int_{R_0} p(\sigma,x,\tau,y)\cdot \frac{1}{\rho^l(y)}\,dy, \tag{19} \]

\[ \Omega_l(\sigma,x,\tau)=\int_\sigma^\tau ds \int_{R_0} p(\sigma,x,s,y)\cdot \frac{1}{\rho^l(y)}\,dy, \tag{20} \]

where \(l<n-k\).

Let the point \(x\) belong to the surface \(\rho(x)=B_9\varepsilon\), where \(B_9>0\). Analogously to how this was done in [3], it is not difficult to obtain the estimates

\[ \omega_l(\sigma,x,\tau)<\frac{\delta(\varepsilon)}{\varepsilon^l} \quad \text{for } \tau-\sigma>\varepsilon, \tag{21} \]

\[ \omega_l(\sigma,x,\tau)<\frac{B_{10}}{\varepsilon^l} \quad \text{for } \tau-\sigma<\varepsilon, \tag{22} \]

\[ \Omega_l(\sigma,x,\tau)<\frac{B_{11}|\ln\varepsilon|}{\varepsilon^{l-2}}, \tag{23} \]

where \(\delta(\varepsilon)\to 0\) as \(\varepsilon\to 0\). One can also obtain an estimate valid for all \(x\in R_0\),

\[ \omega_l(\sigma,x,\tau)\leqslant \frac{B_{12}}{\rho^l(x)} \tag{24} \]

and an estimate valid when \(\rho(x)\geqslant \mathrm{const}>0\),

\[ \Omega_l(\sigma,x,\tau)\leqslant B_{13}. \tag{25} \]

For points \(x\) belonging to the surface \(V(M)\), the inequality

\[ \gamma\varepsilon \leqslant \rho(x)\leqslant \mu\varepsilon, \tag{26} \]

holds, where \(\gamma,\mu\) are positive constants.

Recalling the estimates (14), (16), (18) and using the inequalities (24)—(26), we easily find that when \(\rho(x)\geqslant \mathrm{const}>0\) the terms in formula (17) containing the function \(\Pi(x)\) have order \(o(\varepsilon^{\,n-k-2})\). Thus, the function

\[ \psi(\sigma,x,\tau)= \frac{A(x)}{\rho^{\,n-k-2}(x)}\,\varepsilon^{\,n-k-2} -\int_{R_\varepsilon} q_\varepsilon(\sigma,x,\tau,y)\cdot \frac{A(y)\cdot \varepsilon^{\,n-k-2}}{\rho^{\,n-k-2}(y)}\,dy \]
\[ -\int_\sigma^\tau ds\int_{R_\varepsilon} q_\varepsilon(\sigma,x,s,y)\cdot L\left[\frac{A(y)\cdot \varepsilon^{\,n-k-2}}{\rho^{\,n-k-2}(y)}\right]dy, \tag{27} \]

which is a solution of equation (1), approximates the function \(\varphi(\sigma,x,\tau)\) for \(\rho(x)\geqslant \mathrm{const}>0\) with accuracy up to \(o(\varepsilon^{\,n-k-2})\).

Let us estimate the values of the function \(\psi(\sigma,x,\tau)\) for \(x\) belonging to the surface \(S_{\mu\varepsilon}\), defined by the equation \(\rho(x)=\mu\varepsilon\) (see (26)).

Using equality (27) and inequalities (13), (15), (18), (21)—(23), we easily obtain

\[ \left.\psi(\sigma,x,\tau)\right|_{S_{\mu\varepsilon}} = \begin{cases} \dfrac{A(x)}{\mu^{\,n-k-2}}+\delta_1(\varepsilon), & \text{if } \tau-\sigma>\varepsilon,\\[6pt] O(1), & \text{if } \tau-\sigma\leqslant \varepsilon, \end{cases} \tag{28} \]

where \(\delta_1(\varepsilon)\to 0\) as \(\varepsilon\to 0\).

We shall henceforth agree to denote by \(\delta_i(\varepsilon)\) a quantity infinitesimal together with \(\varepsilon\).

For what follows we shall need

Lemma 1. Let \(u\) be the solution of the following boundary-value problem:

\[ \frac{\partial u}{\partial \sigma} = -a^{ij}\frac{\partial^2 u}{\partial x^i\partial x^j} -b^i\frac{\partial u}{\partial x^i} = L(u), \tag{29} \]

\[ u(\tau,x,\tau)=0,\qquad \left|u(\sigma,x,\tau)\right|_{S_{\mu\varepsilon}} \leqslant \begin{cases} B_{14}, & \text{if } \tau-\sigma\leqslant \varepsilon,\\ \delta_2(\varepsilon), & \text{if } \tau-\sigma>\varepsilon. \end{cases} \tag{30} \]

Then, if \(\rho(x)\geqslant \mathrm{const}>0\),

\[ u(\sigma,x,\tau)=O\!\left(\varepsilon^{\,n-k-2}\right). \]

Proof. Let \(u_1(\sigma,x,\tau)\), \(u_2(\sigma,x,\tau)\) be the solutions of equation (29) with zero initial values and boundary conditions

\[ \left.u_1(\sigma,x,\tau)\right|_{S_{\mu\varepsilon}} = \begin{cases} B_{14}, & \text{if } \tau-\sigma\leqslant \varepsilon,\\ 0, & \text{if } \tau-\sigma>\varepsilon, \end{cases} \qquad \left.u_2(\sigma,x,\tau)\right|_{S_{\mu\varepsilon}} = \delta_2(\varepsilon). \]

By the maximum principle for parabolic equations we have

\[ |u(\sigma,x,\tau)| \leqslant u_1(\sigma,x,\tau)+u_2(\sigma,x,\tau) = u_1(\sigma,x,\tau)+\delta_2(\varepsilon)\chi(\sigma,x,\tau), \tag{31} \]

where \(\chi(\sigma,x,\tau)\) is the solution of equation (29) satisfying the conditions

\[ \chi(\tau,x,\tau)=0,\qquad \left.\chi(\sigma,x,\tau)\right|_{S_{\mu\varepsilon}}=1. \]

The function \(\chi(\sigma,x,\tau)\) can be written in the form of a formula analogous to (17) for \(\varphi(\sigma,x,\tau)\); moreover, the function analogous to \(\Pi(x)\) is equal to 0, since for the solution of the exterior Dirichlet problem on a sphere with unit potential on it, no potential is needed. From this explicit formula and estimates of the type (13), (15), (24), (25), for \(\rho(x)\geqslant \mathrm{const}>0\) we obtain
\(\chi(\sigma,x,\tau)=O(\varepsilon^{\,n-k-2})\). Since \(\delta_2(\varepsilon)\to 0\) as \(\varepsilon\to 0\), it follows that, for \(\rho(x)\geqslant \mathrm{const}>0\),

\[ \delta_2(\varepsilon)\chi(\sigma,x,\tau) = O\!\left(\varepsilon^{\,n-k-2}\right). \tag{32} \]

It remains to estimate the function \(u_1(\sigma,x,\tau)\).

Consider a smooth nonnegative function \(B(x)\), equal to the constant \(B_{14}\) for \(x\in S_{\mu\varepsilon}\) and vanishing for \(x\) lying outside \(R_0\). Now one can define, in the whole space \(R^n\), the function

\[ \gamma(x)= \begin{cases} \dfrac{B(x)}{\rho^{\,n-k-1}(x)}\cdot \varepsilon^{\,n-k-1}, & \text{if } x\in R_0,\\[6pt] 0, & \text{if } x\notin R_0. \end{cases} \tag{33} \]

Denote by \(q_{\mu\varepsilon}(\sigma,x,\tau,y)\) the Green’s function for the exterior boundary-value problem on the surface \(S_{\mu\varepsilon}\) for equation (29). Let

\[ v(\sigma,x,\tau)=\gamma(x)+v_0(\sigma,x,\tau), \tag{34} \]

where

\[ v_0(\sigma,x,\tau) = -\int_{\sigma}^{\tau} ds \int_{R_{\mu\varepsilon}} q_{\mu\varepsilon}(\sigma,x,s,y)\cdot L[\gamma(y)]\,dy, \]

and \(R_{\mu\varepsilon}\) is the set of points \(x \in R_0\) lying outside \(S_{\mu\varepsilon}\). It is clear that

\[ u_1(\sigma,x,\tau)\leqslant v(\sigma,x,\tau). \tag{35} \]

Further,

\[ q_{\mu\varepsilon}(\sigma,x,\tau,y)\leqslant p(\sigma,x,\tau,y), \tag{36} \]

\[ |L[\gamma(x)]|\leqslant \begin{cases} \dfrac{B_{15}\varepsilon^{\,n-k-1}}{\rho^{\,n-k+1}(x)}, & \text{for } x\in R_0,\\[6pt] 0, & \text{for } x\notin R_0. \end{cases} \]

Therefore

\[ |v_0(\sigma,x,\tau)| \leqslant \varepsilon^{\,n-k-1}B_{16} \int_{\sigma}^{\tau} ds \int_{R_{\mu\varepsilon}} p(\sigma,x,s,y)\cdot \frac{dy}{\rho^{\,n-k+1}(y)} \leqslant \]

\[ \leqslant \varepsilon^{\,n-k-2-\nu}B_{17} \int_{\sigma}^{\tau} ds \int_{R_0} p(\sigma,x,s,y)\cdot \frac{dy}{\rho^{\,n-k-\nu}(y)}, \tag{37} \]

where \(\nu\) is any constant satisfying \(0<\nu<1\). Applying inequality (24) to the inner integral and taking \(\tau-\sigma\leqslant\varepsilon\), we obtain from inequality (37)

\[ |v_0(\sigma,x,\tau)|\leqslant \begin{cases} \dfrac{B_{18}}{\rho^{\,n-k-\nu}(x)} \varepsilon^{\,n-k-1-\nu}, & \text{for } x\in R_0,\\[6pt] B_{19}\varepsilon^{\,n-k-1-\nu}, & \text{for } x\notin R_0. \end{cases} \tag{38} \]

For \(\tau-\sigma>\varepsilon\), the boundary values of \(u_1(\sigma,x,\tau)\) are equal to zero; therefore, for \(\tau-\sigma>\varepsilon\),

\[ u_1(\sigma,x,\tau) = \int_{R^n} q_{\mu\varepsilon}(\sigma,x,\tau-\varepsilon,y)\cdot u_1(\tau-\varepsilon,y,\tau)\,dy. \tag{39} \]

From relations (34), (35), (36), (38), (39), for \(\tau-\sigma>\varepsilon\) we obtain

\[ u_1(\sigma,x,\tau) \leqslant \varepsilon^{\,n-k-1}B_{20} \int_{R_0} p(\sigma,x,\tau_1,y)\cdot \frac{dy}{\rho^{\,n-k-1}(y)} + \]

\[ + \varepsilon^{\,n-k-1-\nu}B_{21} \int_{R_0} p(\sigma,x,\tau_1,y)\cdot \frac{dy}{\rho^{\,n-k-\nu}(y)} + B_{19}\varepsilon^{\,n-k-1-\nu}, \]

where \(\tau_1=\tau-\varepsilon\). Applying inequality (24), hence, for \(\rho(x)\geqslant \mathrm{const}>0\) we obtain

\[ u_1(\sigma,x,\tau)=O(\varepsilon^{\,n-k-2}). \tag{40} \]

From equalities (32), (40) and inequality (31), the validity of the lemma follows.

§ 2. Derivation of the formula for the probability \(\varphi(\sigma,x,\tau)\). The case of constant coefficients

In this paragraph we shall use the preceding constructions and derive a simple formula for the principal term of the probability \(\varphi(\sigma,x,\tau)\).

Denote by \(\widetilde w(\xi^{k+1},\ldots,\xi^n)\) the harmonic function tending to \(0\) as \(|\xi|\to+\infty\) and equal to unity on the ellipsoid \(\widetilde E_m\) singled out in \(Q(m)\) by the equation (compare with equation (3) for the ellipsoid \(E_m\))

\[ \sum_{i,j=k+1}^{n} c_{ij}(m)\xi^i\xi^j=1. \]

It is known that \(\widetilde w\) can be represented in the form

\[ \widetilde w=\frac{\alpha(m)}{r^{\,n-k-2}(\xi)}+\widetilde\pi_m(\xi^{k+1},\ldots,\xi^n), \tag{41} \]

where \(r^2(\xi)=(\xi^{k+1})^2+\cdots+(\xi^n)^2\); \(\alpha(m)\) coincides with \(\alpha(m)\) in equality (5); \(\widetilde\pi_m\) is the double-layer potential created by the ellipsoid \(\widetilde E_m\) at the point \((\xi^{k+1},\ldots,\xi^n)\).

Differentiating the right- and left-hand sides of equality (41) in the direction of the exterior normal to \(\widetilde E_m\) and then taking the integral over the surface \(\widetilde E_m\), we find that

\[ \int_{\widetilde E_m}\frac{\partial \widetilde w}{\partial N}\,d\widetilde E_m = \frac{4\pi^{\frac{n-k}{2}}}{\Gamma\!\left(\frac{n-k}{2}-1\right)}\,\alpha(m) = \beta(m), \tag{42} \]

where \(\Gamma\) is Euler’s gamma function; \(N\) is the exterior normal to \(\widetilde E_m\).

Let \(a=\det\|a^{ij}\|\). We can now formulate the principal result of the present paper:

The solution of equation (1) under the conditions (2) can be represented in the form

\[ \varphi(\sigma,x,\tau) = \varepsilon^{\,n-k-2} \int_{\sigma}^{\tau} ds \int_M p(\sigma,x,s,m)\,\beta(m)\,\sqrt{a}\,dM - \omega(\sigma,x,\tau,\varepsilon), \tag{43} \]

where \(\omega\) is of order \(o(\varepsilon^{\,n-k-2})\) for any point \(x\) separated from the manifold \(M\) by a finite distance independent of \(\varepsilon\).

In formula (43) the inner integration is carried out over the entire manifold \(M\), the volume element in which is induced at each point by the frame \(e_1,\ldots,e_k\). This definition of volume depends only on the coefficients \(a^{ij}\) of equation (1) and does not depend on the permissible arbitrariness in the choice of the frame \(e_1,\ldots,e_k\).

To prove the assertion just formulated, we shall estimate the boundary values of the function

\[ \Phi(\sigma,x,\tau) = \varepsilon^{\,n-k-2} \int_{\sigma}^{\tau} ds \int_M p(\sigma,x,s,m)\,\beta(m)\,\sqrt{a}\,dM \tag{44} \]

on the surface \(S_{\mu\varepsilon}\), defined by the equality \(\rho(x)=\mu\varepsilon\).

Denote by \(m_0\) the projection of the point \(x\) onto the manifold \(M\) in the direction of the plane \(Q(m_0)\). With the point \(m_0\) there is associated the linear coordinate transformation

\[ y=m_0+A(m_0)\eta, \tag{45} \]

which transforms the operator \(a^{ij}\dfrac{\partial^2}{\partial y^i \partial y^j}\) into the Laplace operator, and such that the first \(k\) columns of the matrix \(A(m_0)\) are vectors parallel to the tangent plane \(P(m_0)\).

We shall agree to understand by \(\tilde \eta\) the vector \((\eta^1,\ldots,\eta^k)\). Obviously, there exists a positive constant \(r_1\) such that, for

\[ r(\tilde \eta)=\sqrt{(\eta^1)^2+\cdots+(\eta^k)^2}\leqslant r_1 \tag{46} \]

the manifold \(M\) can be written in the form

\[ \eta^{k+1}=\varphi_{k+1}(\tilde \eta), \]

\[ \cdot \quad \cdot \quad \cdot \quad \cdot \tag{47} \]

\[ \eta^n=\varphi_n(\tilde \eta), \]

where the functions \(\varphi_{k+1},\ldots,\varphi_n\) are twice continuously differentiable. By virtue of the special choice of the coordinate system, the functions \(\varphi_{k+1},\ldots,\varphi_n\) and their first partial derivatives at \(\tilde \eta=0\) are equal to \(0\); therefore, expanding these functions in Taylor series with center at the point \(\tilde \eta=0\) up to and including terms of second order, we obtain, for \(r(\tilde \eta)\leqslant r_1\),

\[ \eta^{k+1}=O(r^2(\tilde \eta)), \]

\[ \cdot \quad \cdot \quad \cdot \quad \cdot \tag{48} \]

\[ \eta^n=O(r^2(\tilde \eta)). \]

The cylinder \(r(\tilde \eta)=r_1\) (see (46)) cuts out from the manifold \(M\) a piece \(M_1\). Put \(M_2=M-M_1\). On \(M_2\), \(|x-m|\geqslant \mathrm{const}>0\); therefore, for \(m\in M_2\), \(p(\sigma,x,s,m)\leqslant \mathrm{const}\), and consequently

\[ J_1=\varepsilon^{\,n-k-2}\int_{\sigma}^{\tau} ds \int_{M_2} p(\sigma,x,s,m)\cdot \beta(m)\cdot \sqrt{a}\,dM =O(\varepsilon^{\,n-k-2}). \tag{49} \]

Let us estimate the integral

\[ J_2=\varepsilon^{\,n-k-2}\int_{\sigma}^{\tau} ds \int_{M_1} p(\sigma,x,s,m)\cdot \beta(m)\cdot \sqrt{a}\,dM \tag{50} \]

for \(x\in S_{\mu\varepsilon}\). Put (see (45))

\[ x=m_0+A(m_0)\xi, \tag{51} \]

\[ y=m_0+A(m_0)\eta \tag{52} \]

and denote

\[ p^*(\sigma,\xi,s,\eta) = p\bigl(\sigma,m_0+A(m_0)\xi,s,m_0+A(m_0)\eta\bigr)\sqrt{a}. \]

Formulas (52), (47) give a parametrization of the surface \(M_1\). The point \(x\in S_{\mu\varepsilon}\) is transformed by (51) into the point

\[ \xi=(0,\ldots,0,\xi^{k+1},\ldots,\xi^n), \tag{53} \]

where

\[ (\xi^{k+1})^2+\cdots+(\xi^n)^2=\mu^2\varepsilon^2. \]

Let \(\eta < M_1\). Using relations (48), (53) and the known representation of the fundamental solution \(p^*(\sigma,\xi,s,\eta)\) (see [6]), we obtain

\[ p^*(\sigma,\xi,s,\eta) = \frac{e^{-\frac{f}{4(s-\sigma)}}}{[4\pi(s-\sigma)]^{\frac n2}} + O\left( \frac{e^{-\gamma \frac{f}{s-\sigma}}}{(s-\sigma)^{\frac{n-\lambda}{2}}} \right), \tag{54} \]

where

\[ f=r^2(\tilde{\eta})\bigl(1+O(\varepsilon)+O(r^2(\tilde{\eta}))\bigr)+\mu^2\varepsilon^2; \tag{55} \]

\(\gamma\) and \(\lambda\) are positive constants. We note that

\[ dM=D(\tilde{\eta})\,d\tilde{\eta}, \tag{56} \]

where \(D(\tilde{\eta})=D(0)+O(r(\tilde{\eta}))=1+O(r(\tilde{\eta}))\).

Thus, from equalities (50), (54), (56) it follows that

\[ \begin{aligned} J_2 ={}& \varepsilon^{\,n-k-2}\beta(m_0) \int_{\sigma}^{\tau} ds \int_{r(\tilde{\eta})<r_1} \frac{e^{-\frac{f}{4(s-\sigma)}}}{[4\pi(s-\sigma)]^{\frac n2}}\,d\tilde{\eta} \\ &+ \varepsilon^{\,n-k-2} \int_{\sigma}^{\tau} ds \int_{r(\tilde{\eta})<r_1} \frac{e^{-\frac{f}{4(s-\sigma)}}}{[4\pi(s-\sigma)]^{\frac n2}}\, O(r(\tilde{\eta}))\,d\tilde{\eta} \\ &+ \varepsilon^{\,n-k-2} \int_{\sigma}^{\tau} ds \int_{r(\tilde{\eta})<r_1} O\left( \frac{e^{-\gamma\frac{f}{s-\sigma}}}{(s-\sigma)^{\frac{n-\lambda}{2}}} \right)\,d\tilde{\eta} = J_3+J_4+J_5 . \end{aligned} \tag{57} \]

We shall now show that the integral \(J_3\) gives a finite quantity, while the integrals \(J_4, J_5\) are infinitely small together with \(\varepsilon\).

Put

\[ q=\frac{n-k-2-\nu}{2n}, \]

where \(0<\nu=\mathrm{const}<1\). We split the integral \(J_3\) into two summands:

\[ \begin{aligned} J_3 ={}& \varepsilon^{\,n-k-2}\beta(m_0) \int_{\sigma}^{\tau} ds \int_{r(\tilde{\eta})<\varepsilon^q} \frac{e^{-\frac{f}{4(s-\sigma)}}}{[4\pi(s-\sigma)]^{\frac n2}}\,d\tilde{\eta} \\ &+ \varepsilon^{\,n-k-2}\beta(m_0) \int_{\sigma}^{\tau} ds \int_{\varepsilon^q<r(\tilde{\eta})<r_1} \frac{e^{-\frac{f}{4(s-\sigma)}}}{[4\pi(s-\sigma)]^{\frac n2}}\,d\tilde{\eta} = J_6+J_7 . \end{aligned} \tag{58} \]

Recalling formula (55) and taking \(r_1\) sufficiently small, we have

\[ J_7 \leq B_{22}\varepsilon^{\,n-k-2}\int_{\sigma}^{\tau} ds \int_{\varepsilon^q< r(\tilde{\eta})\leq r_1} \frac{e^{-B_{23}\cdot \frac{\varepsilon^{2q}}{s-\sigma}}}{(s-\sigma)^{n/2}}\,d\tilde{\eta} \leq B_{24}\varepsilon^{\,n-k-2}\int_{\sigma}^{\tau} \frac{e^{-B_{23}\cdot \frac{\varepsilon^{2q}}{s-\sigma}}}{(s-\sigma)^{n/2}}\,ds . \tag{59} \]

Applying the elementary inequality

\[ v^{B_{25}}e^{-B_{26}v}\leq B_{27}e^{-B_{28}v} \quad (0\leq v<+\infty,\; 0\leq B_{25},\; 0<B_{28}<B_{26}), \]

we continue inequality (59):

\[ J_7 \leq B_{29}\frac{\varepsilon^{\,n-k-2}}{\varepsilon^{nq}} \leq B_{29}\varepsilon^{\nu/2}. \tag{60} \]

Consider the integral \(J_6\). For it the function \(f\) (see formula (55)) is estimated as follows:

\[ r^2(\tilde{\eta})(1-B_{30}\varepsilon^{2q})+\mu^2\varepsilon^2 \leq f \leq r^2(\tilde{\eta})(1+B_{31}\varepsilon^{2q})+\mu^2\varepsilon^2 . \tag{61} \]

Therefore

\[ \begin{aligned} J_6 &\leq \varepsilon^{\,n-k-2}\beta(m_0) \int_{\sigma}^{\tau} ds\, \frac{e^{-\frac{\mu^2\varepsilon^2}{4(s-\sigma)}}}{[4\pi(s-\sigma)]^{(n-k)/2}} \int_{R^k} \frac{e^{-\frac{r^2(\tilde{\eta})(1-B_{30}\varepsilon^{2q})}{4(s-\sigma)}}}{[4\pi(s-\sigma)]^{k/2}}\,d\tilde{\eta} \\ &= \varepsilon^{\,n-k-2}\beta(m_0)(1+\delta_3(\varepsilon)) \int_{\sigma}^{\tau} \frac{e^{-\frac{\mu^2\varepsilon^2}{4(s-\sigma)}}}{[4\pi(s-\sigma)]^{(n-k)/2}}\,ds . \end{aligned} \]

In paper [4] such an integral is evaluated; we obtain

\[ J_6 \leq \begin{cases} O(1), & \text{for } \tau-\sigma\leq \varepsilon,\\[4pt] \dfrac{\alpha(m_0)}{\mu^{\,n-k-2}}+\delta_4(\varepsilon), & \text{for } \tau-\sigma>\varepsilon . \end{cases} \tag{62} \]

Starting from inequality (61), in an analogous way we show that

\[ J_6 \geq \begin{cases} O(1), & \text{for } \tau-\sigma\leq \varepsilon,\\[4pt] \dfrac{\alpha(m_0)}{\mu^{\,n-k-2}}+\delta_5(\varepsilon), & \text{for } \tau-\sigma>\varepsilon . \end{cases} \tag{63} \]

From inequalities (60), (62), (63) we obtain

\[ J_3 = \begin{cases} O(1), & \text{for } \tau-\sigma\leq \varepsilon,\\[4pt] \dfrac{\alpha(m_0)}{\mu^{\,n-k-2}}+\delta_6(\varepsilon), & \text{for } \tau-\sigma>\varepsilon . \end{cases} \]

The integrals \(J_4, J_5\) (see equality (57)) are estimated in the same way as the integral \(J_3\). In view of the fact that \(J_4\) contains \(O(r(\eta))\), while \(J_5\) contains \((s-\sigma)\) to the power \(\dfrac{n-k}{2}\), they are infinitely small together with \(\varepsilon\).

Thus, we have obtained

\[ \left. \Phi(\sigma,x,\tau)\right|_{S_{\mu\varepsilon}} = \left| \begin{array}{ll} O(1), & \text{for } \tau-\sigma < \varepsilon,\\[6pt] \dfrac{\alpha(m_0)}{\rho^{\,n-k-2}}+\delta_7(\varepsilon), & \text{for } \tau-\sigma > \varepsilon . \end{array} \right. \tag{64} \]

It is easy to see that the function \(\Phi(\sigma,x,\tau)\) is a solution of equation (1) outside the manifold \(M\). From relations (28), (64), according to Lemma 1, we obtain that, for \(\rho(x) \geq \mathrm{const}>0\), it approximates the function \(\psi(\sigma,x,\tau)\), and hence also \(\varphi(\sigma,x,\tau)\), with accuracy up to \(o(\varepsilon^{\,n-k-2})\).

Thus formula (43) is proved.

§ 3. DERIVATION OF THE FORMULA FOR THE PROBABILITY \(\varphi(\sigma,x,\tau)\). THE CASE OF VARIABLE COEFFICIENTS

Consider the case when the coefficients of equation (1) are variable, and the manifold \(M\) changes its form and position according to the law \(M=M_s\).

Through each point \(m_s\in M_s\) we draw the tangent plane \(P(m_s)\). We then choose \(n\) linearly independent vectors \(e_1,\ldots,e_n\), issuing from the point \(m_s\), so that: a) \(e_1,\ldots,e_k\) belong to \(P(m_s)\); b) in the coordinate system \(\xi^1,\ldots,\xi^n\), referred to the basis \(e_1,\ldots,e_n\), the differential operator
\[ a^{ij}(s,m_s)\frac{\partial^2}{\partial x^i\partial x^j} \]
is written in the form of the Laplace operator
\[ \sum_{\nu=1}^{n}\frac{\partial^2}{(\partial \xi^\nu)^2}. \]

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 distance \(\varepsilon\) from the plane \(P(m_s)\) in the metric \(R^n\) is an ellipsoid \(E_{m_s}^{\varepsilon}\). Its equation in the coordinates \(\xi\) shall be
\[ \sum_{i,j=k+1}^{n} c_{ij}(m_s)\xi^i\xi^j=\varepsilon^2. \]

Obviously, with accuracy up to small quantities of higher order in \(\varepsilon\), we have \(V(M_s)=E_{m_s}^{\varepsilon}\times M_s\).

Denote by \(\widetilde{w}(\xi^{k+1},\ldots,\xi^n)\) the harmonic function tending to \(0\) as \(|\xi|\to+\infty\) and equal to one on the ellipsoid \(\widetilde{E}_{m_s}\) defined in \(Q(m_s)\) by the equation
\[ \sum_{i,j=k+1}^{n} c_{ij}(m_s)\xi^i\xi^j=1. \]

It is known that \(\widetilde{w}\) can be represented in the form

\[ \widetilde{w} = \frac{\alpha(m_s)}{r^{\,n-k-2}(\xi)} + \widetilde{\pi}_{m_s}(\xi^{k+1},\ldots,\xi^n), \tag{65} \]

where \(r^2(\xi)=(\xi^{k+1})^2+\cdots+(\xi^n)^2\); \(\alpha(m_s)\) is uniquely determined by the dimensions of the ellipsoid \(\widetilde{E}_{m_s}\); \(\widetilde{\pi}_{m_s}\) is the double-layer potential generated by the ellipsoid \(\widetilde{E}_{m_s}\).

Differentiating the right- and left-hand sides of relation (65) in the direction of the exterior normal to \(\widetilde E_{m_s}\) and then taking the integral over the surface \(\widetilde E_{m_s}\), we find that

\[ \int_{\widetilde E_{m_s}} \frac{\partial \widetilde w}{\partial N}\, d\widetilde E_{m_s} = \frac{4\pi^{\frac{n-k}{2}}}{\Gamma\!\left(\frac{n-k}{2}-1\right)}\,\alpha(m_s)-\beta(m_s), \]

where \(\Gamma\) is Euler’s gamma function; \(N\) is the exterior normal to \(\widetilde E_{m_s}\).

Let \(a(s,m_s)\) be the determinant of the matrix \(a^{ij}(s,m_s)\). We can now formulate the following proposition:

The solution of equation (1) under conditions (2) can be represented in the form

\[ \varphi(\sigma,x,\tau) = \varepsilon^{\,n-k-2} \int_{\sigma}^{\tau} ds \int_{M_s} p(\sigma,x,s,m_s)\,\beta(m_s)\,\sqrt{a(s,m_s)}\,dM_s + \omega(\sigma,x,\tau,\varepsilon), \tag{66} \]

where \(\omega\) is of order \(o(\varepsilon^{\,n-k-2})\) for any point \(x\) whose distance from the manifold \(M_\sigma\) is finite and independent of \(\varepsilon\).

In formula (66), the inner integration is performed over the entire manifold \(M_s\), the volume element in which is induced at each point by the frame \(e_1,\ldots,e_k\). This definition of volume depends only on the coefficients \(a^{ij}\) of equation (1) and does not depend on the admissible arbitrariness in the choice of the frame \(e_1,\ldots,e_k\).

Formula (66) is proved in the same way as (43).

§ 4. DERIVATION OF THE FORMULA FOR THE PROBABILITY \(\varphi(\sigma,x,\tau)\) FOR \(n-k=2\)

We now consider the case \(n-k=2\).

Here the following formula turns out to be valid:

\[ \varphi(\sigma,x,\tau) = \frac{2\pi}{|\ln\varepsilon|} \int_{\sigma}^{\tau} ds \int_{M_s} p(\sigma,x,s,m_s)\,\sqrt{a(s,m_s)}\,dM_s + \omega(\sigma,x,\tau,\varepsilon), \tag{67} \]

where \(a(s,m_s)=\det\|a^{ij}(s,m_s)\|\); \(\omega\) is of order \(o\!\left(\frac{1}{|\ln\varepsilon|}\right)\) for any point \(x\) whose distance from the manifold \(M_\sigma\) is finite and independent of \(\varepsilon\).

The volume element \(dM_s\) is defined in the same way as for \(n-k\ge 3\).

The proof of formula (67) is carried out analogously to the proof given in [5]. The scheme of the proof is as follows.

The function

\[ \Phi(\sigma,x,\tau) = \frac{2\pi}{|\ln\varepsilon|} \int_{\sigma}^{\tau} ds \int_{M_s} p(\sigma,x,s,m_s)\,\sqrt{a(s,m_s)}\,dM_s \]

is a solution of equation (1) outside the manifold \(M_\sigma\). Note that for points of the surface \(V(M_\sigma)\), defined by equation (4), it is valid—

or the inequality \(0<\mu_2\varepsilon<\rho_\sigma(x)<\mu_1\varepsilon\), where \(\mu_1,\mu_2\) are constants. Denote by \(S_{\mu_1\varepsilon}\) the surface \(\rho_\sigma(x)=\mu_1\varepsilon\), and by \(S_{\mu_2\varepsilon}\) the surface \(\rho_\sigma(x)=\mu_2\varepsilon\). In the same way as was done for \(n-k\ge 3\), it is shown that

\[ \left.\Phi(\sigma,x,\tau)\right|_{S_{\mu_1\varepsilon}} = \begin{cases} O(1), & \text{for } \tau-\sigma<|\ln\varepsilon|^{-5},\\ 1+\delta_8(\varepsilon), & \text{for } \tau-\sigma>|\ln\varepsilon|^{-5}, \end{cases} \]

and

\[ \left.\Phi(\sigma,x,\tau)\right|_{S_{\mu_2\varepsilon}} = \begin{cases} O(1), & \text{for } \tau-\sigma<|\ln\varepsilon|^{-5},\\ 1+\delta_9(\varepsilon), & \text{for } \tau-\sigma>|\ln\varepsilon|^{-5}. \end{cases} \]

By the maximum principle for parabolic equations we obtain

\[ \left.\Phi(\sigma,x,\tau)\right|_{V(M_\sigma)} = \begin{cases} O(1), & \text{for } \tau-\sigma<|\ln\varepsilon|^{-5},\\ 1+\delta_{10}(\varepsilon), & \text{for } \tau-\sigma>|\ln\varepsilon|^{-5}. \end{cases} \tag{68} \]

We next prove a lemma.

Lemma 2. Let \(u\) be the solution of the following boundary-value problem:

\[ \frac{\partial u}{\partial \sigma} = -a^{ij}(\sigma,x)\frac{\partial^2 u}{\partial x^i\partial x^j} -b^i(\sigma,x)\frac{\partial u}{\partial x^i} =L(u), \]

\[ u(\tau,x,\tau)=0,\qquad |u(\sigma,x,\tau)|\big|_{V(M_\sigma)} < \begin{cases} B_{32}, & \text{for } \tau-\sigma<|\ln\varepsilon|^{-5},\\ \delta_{11}(\varepsilon), & \text{for } \tau-\sigma>|\ln\varepsilon|^{-5}. \end{cases} \tag{69} \]

Then for the solution \(u(\sigma,x,\tau)\), for \(\rho_\sigma(x)>\operatorname{const}>0\), the estimate

\[ |u(\sigma,x,\tau)|<\delta_{11}(\varepsilon)\chi(\sigma,x,\tau)+o\!\left(\frac{1}{|\ln\varepsilon|}\right) \]

is valid, where \(\chi(\sigma,x,\tau)\) is the solution of equation (69), satisfying the zero initial condition and assuming the value one on \(V(M_\sigma)\).

From equality (68) and this lemma it follows that formula (67) is valid.

In conclusion, we want to note one particular case of the manifold \(M_s\) (\(n-k\ge 2\)). Namely, suppose that in the \(n\)-dimensional Euclidean space \(R^n\) of variables \(z^1,\ldots,z^n\) it is given by the equations

\[ z^1=z^1, \]

\[ \cdots \]

\[ z^k=z^k, \]

\[ z^{k+1}=z^{k+1}(s), \]

\[ \cdots \]

\[ z^n=z^n(s), \]

where \(z^{k+1}(s),\ldots,z^n(s)\) are continuously differentiable functions. Then the surface \(V(M_s)\) is the cylinder

\[ \bigl(z^{k+1}-z^{k+1}(s)\bigr)^2+\cdots+\bigl(z^n-z^n(s)\bigr)^2=\varepsilon^2. \]

The probability of hitting the \(\varepsilon\)-neighborhood of such a manifold can be interpreted as the probability of agreement, with accuracy up to \(\varepsilon\), in the coordinates \(z^{k+1},\ldots,z^n\) of a random point \(x\) with some point \(z(s)=(z^1(s),\ldots,z^{k+1}(s),\ldots,z^n(s))\) moving deterministically. This problem is a direct generalization of the problem posed in [3].

References

1. Mishchenko E. F. DAN SSSR, 159, 2, 266—268, 1964.

2. Kolmogorov A. N. Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung. Math. Ann., 415—458, 1931.

3. Mishchenko E. F., Pontryagin L. S. Izv. AN SSSR, ser. matem., 25, 477—498, 1961.

4. Kolmogorov A. N., Mishchenko E. F., Pontryagin L. S. DAN SSSR, 145, 5, 993—995, 1962.

5. Nikol’skii M. S. Theory of Probability and Its Applications, 9, no. 2, 352—357, 1964.

6. Il’in A. M., Kalashnikov A. S., Oleinik O. A. UMN, vol. XVII, 3, 3—146, 1962.

Received by the editors
January 10, 1966

V. A. Steklov Mathematical Institute