Mathematics

Yu. L. Daletskii

CONTINUAL INTEGRALS ASSOCIATED WITH CERTAIN DIFFERENTIAL EQUATIONS AND SYSTEMS

(Presented by Academician A. N. Kolmogorov, 23 IX 1960)

In the present note, the method set forth in (1) for representing solutions in the form of continual integrals is extended to a new class of differential equations.

1. Let (M(x_0, x_l)) be the space of bounded vector-functions (x(t)) ((0 \leqslant t \leqslant l)) with values in the finite-dimensional space (R_\nu), satisfying the conditions (x(0)=x_0), (x(l)=x_l). Further, let (q) be a partition
   [
   0=t_0<t_1<\cdots<t_n<t_{n+1}=l;
   ]
   (R_\nu^q) the space of elements ((x_1,x_2,\ldots,x_n)), where (x_k\in R_\nu), and (Q(q,R)) the cylindrical set in (M(x_0,x_l)) generated by the partition (q) and the Borel set (R) from (R_\nu^q). Suppose that on the sets (Q(q,R)) a function (\mu(q,R)) is defined, whose values are (m)-dimensional matrices and which is an abstract measure of bounded variation in each (R_\nu^q), satisfying the usual consistency conditions. Such a function (\mu_S(q,R)) is constructed in the usual way (as in the theory of Markov processes) if a family of matrices (S(t_1,t_2;x_1,x_2)) ((t_1<t_2)) is given, satisfying, for some measure (\sigma(x)) in (R_\nu), the equation

[
\int_{R_\nu} S(t_2,t_3;x_2,x_3)S(t_1,t_2;x_1,x_2)\,d\sigma(x_2)
= S(t_1,t_3;x_1,x_3).
\tag{1}
]

If (\Phi(x(t))) is a functional on (M(x_0,x_l)), then put

[
I(\Phi)=\lim_q I_q(\Phi)=\lim_q \int_{R_\nu^q} \Phi(x_q(t))\,\mu(q,dx),
\tag{2}
]

where (x_q(t)=x(t_{k-1})) for (t_{k-1}\leqslant t<t_k) ((k=1,\ldots,n+1)) and (x_q(t)=x_l) for (t=l). If (I(\Phi)) has meaning, then we shall call it a continual integral and shall denote it by

[
I(\Phi)=\int_{M(x_0,x_l)}^{*} \Phi(x(t))\,d\mu(x(t)).
\tag{3}
]

If the function (\mu(q,R)) has bounded variation, then it can be extended in the usual way to a certain countably additive measure. Integrals with respect to this measure, in contrast to (3), will be written without the asterisk. For bounded continuous functionals (\Phi(x)), the two definitions coincide (2).

Theorem 1. If (S(t_1,t_2;x_1,x_2)) is a real function continuous in (x_1,x_2) and
[
\int_{R_\nu} S(t_1,t_2;x_1,x_2)\,d\sigma(x_2)\geqslant 1,
]
then the function (\mu_S(q,R)) has bounded variation if and only if, in the formula written above, the equality sign holds and (S(t_1,t_2;x_1,x_2)\geqslant 0).

1. Let (\mathfrak H) be a Hilbert space; (T) a self-adjoint operator with bounded inverse; (D=D_T) the space of fundamental elements constructed from (T); (N) the corresponding space of generalized elements (1).

Consider the solution (U(\tau,t)) of the operator equation (dU/dt=A(t)U) with the condition (U(\tau,\tau)=I), assuming that the conditions under which it exists and is well-defined are satisfied ((^{3-6})). Suppose that for some (p>0), (|TA^{-p}|<\infty). Then the expressions
[
S_{jk}(t_1,t_2;x_1,x_2)=\bigl(U(t_1,t_2)\xi_{jx_1},\xi_{kx_2}\bigr),
]
where (\xi_{kx}) ((k=1,\ldots,m;\ x\in R_\nu)) is a complete family of generalized elements, are meaningful, and the matrix (S=|S_{jk}|) satisfies condition (1).

Theorem 2. Let (m=1) and
[
S(t_1,t_2;x_1,x_2)=\bigl(e^{A(t_2-t_1)}\xi_{x_1},\xi_{x_2}\bigr)
]
((A=\mathrm{const})). Suppose there exists a sequence (\varphi_n\in D) such that (A\varphi_n\in D), and the sequences of functions (\varphi_n(x)=(\varphi_n,\xi_x)) and (\psi_n(x)=(A\varphi_n,\xi_x)) are uniformly bounded, with (\varphi_n(x)\to 1), (\psi_n(x)\to 0) as (n\to\infty) almost everywhere in (R_\nu) with respect to (\sigma(x)). Then, if
[
\int_{R_\nu}|S(t_1,t_2;x_1,x_2)|\,d\sigma(x_2)<\infty
]
for (t_2>t_1), then
[
\int_{R_\nu} S(t_1,t_2;x_1,x_2)\,d\sigma(x_2)=1.
]

An analogous theorem also holds for (m>1) and for (A) depending on (t).

1. Let (A) be a self-adjoint negative definite operator, and let (B) be some generating operator such that (A+B) is the generating operator of a semigroup ((^6)) that maps (\mathfrak H) into (D). In ((^1)), under the boundedness condition on the operators (B) and (A^pBA^{-p}), the equality
   [
   e^{(A+B)t}\xi=\lim_q \sum_{k=1}^n e^{A(t_k-t_{k-1})}e^{B(t_k-t_{k-1})}\xi
   \quad(\xi\in N),
   \tag{4}
   ]
   is established, where the limit is understood in the sense of strong convergence in (D).

Theorem 3. Relation (4) is valid if, for some (\gamma>0) and (p_1\ge p), the conditions
[
|A^pBA^{-p_1}|<\infty,\qquad
|A^{p_1}e^{Bt}A^{-p_1}|\le e^{\gamma t}
]
are satisfied.

Let us introduce the fundamental matrix of the operator equation (d\psi/dt=(A+B)\psi), corresponding to the chosen complete system of generalized elements:
[
W(t;x,y)=\bigl|\bigl(e^{(A+B)t}\xi_{ix},\xi_{jy}\bigr)\bigr|_{i,j=1}^m .
]

From formula (4) follows the representation
[
W(t;x_0,x_l)=\lim_q \int_{R_\nu^q}\sum_{r=1}^{n+1} G(t_r-t_{r-1};x_{r-1},x_r)\,d\sigma(x_1)\cdots d\sigma(x_n),
\tag{5}
]
where
[
G_{ij}(t;x,y)=\bigl(e^{At}e^{Bt}\xi_{ix},\xi_{jy}\bigr).
]
We note that these results also carry over to the case when the operators (A) and (B) depend on (t).

1. Let us consider some special cases.

1) Let (B=V(C)) be a bounded scalar function of a self-adjoint operator (C), satisfying conditions similar to those considered in (1). Under these conditions, as is known ((^7)), (N) contains a complete family of generalized eigen-elements of the operator (C). From formula (5), for the fundamental matrix corresponding to this system, there follows the representation
[
W(t;x_0,x_l)=
\int_{M(x_0,x_l)}^{*}
\exp\left[\int_0^t V(x(t))\,dt\right]\,d\mu_S(x(t)),
\tag{6}
]
where
[
S_{ij}(t;x,y)=\bigl(e^{At}\xi_{ix},\xi_{jy}\bigr).
]
Formula (6) can be used to represent the fundamental matrix of the system
[
\partial\psi/\partial t=L(\psi)+B(x,t)\psi,
\tag{7}
]

where (\psi) is an (m)-dimensional vector, (L(\psi)) is a self-adjoint strongly elliptic system of differential operators, and (B(x,t)) is a Hermitian matrix.

2) Let
[
B\xi_{jx}=\sum_{k=1}^{m}\beta_{jk}(x)\xi_{kx}
]
and
[
(e^{At}\xi_{jx},\xi_{ky})=\delta_{jk}(e^{At}\xi_{1x},\xi_{1y}).
]
Then
[
W(l;x_0,x_l)=\int_{M(x_0,x_l)}^{*}\exp\left[\int_{0}^{l}\beta(x(t))\,dt\right]d\mu_S(x(t)),
]
where
[
S(t;x,y)=(e^{At}\xi_{1x},\xi_{1y})
]
is a scalar function, and under the sign of the continual integral there stands a multiplicative integral. This formula can be applied to a system of the form (7), in which (L(\psi)=L_1(\psi)\cdot I), where (L_1(\psi)) is an elliptic operator and (I) is the identity matrix.

3) For simplicity consider the case (m=1), and let
[
e^{Bt}\xi_x=e^{t\beta(x,t)}\xi_{\gamma(x,t)},
]
where (\beta(x,t)) is a scalar function and (\gamma(x,t)) is a function with values in (R_\nu). Formula (5) can then be brought to the form
[
W(l;x_0,x_l)=
]
[
=\lim_b\int_{R_q^\nu}\exp\left{\sum_{r=1}^{n+1}\left[\Delta t_r\beta(x_{r-1},\Delta t_r)+f(\Delta t_r;x_{r-1},x_r)\right]\right}d\mu_S(q,R).
\tag{8}
]
where
[
S(t;x,y)=(e^{At}\xi_x,\xi_y)
]
and
[
f(t;x,y)=\ln\frac{S(t,\gamma(x,y),y)}{S(t,x,y)}.
]

Let there exist a functional (\Phi(x(t))) in (M(x_0,x_l)) for which the value (\Phi(x_q(t))) coincides with the expression under the sign (\exp) in (8). Then
[
W(l;x_0,x_l)=\int_{M(x_0,x_l)}^{*}e^{\Phi(x(t))}d\mu(x(t)).
\tag{9}
]

Carrying out analogous considerations in the case (m\geqslant1), one can obtain a representation of the fundamental matrix of a system of the form
[
\frac{\partial\psi}{\partial t}=L(\psi)\cdot I+\sum_{k=1}^{\nu}a_k(x,t)\frac{\partial\psi}{\partial x^{(k)}}+V(x,t)\psi,
\tag{10}
]
where (I(\psi)) is a self-adjoint elliptic operator, and (a_k(x,t)) and (V(x,t)) are certain matrices.

Theorem 4. Let, in system (10), (L(\psi)=a\Delta\psi) and (a_k=U_k\Lambda_kU_k^{-1}), where (\Lambda_k) are diagonal matrices. Then
[
W(l;x_0,x_l)=
]
[
=\int_{M(x_0,x_l)}^{*}\exp\left{-\frac{1}{2a}\int_{0}^{l}\sum_{k=1}^{\nu}a_k\,dx^{(k)}
-\frac{1}{4a}\int_{0}^{l}\sum_{k=1}^{\nu}(a_k^2-U_k^{1}U_k^{-1}a_k)\,dt
-\int_{0}^{l}\operatorname{div}a(x)\,dt\right}\times
]
[
\times d\mu(x(t)).
\tag{11}
]

In this connection, for a functional of the form
[
\Psi(x(t))=\int_{0}^{l}f(x,\dot t)\,dx(t)
]
we set
[
\Psi(x_q(t))=\sum_{r=1}^{n+1}f(x_{r-1},t_{r-1})(x_r-x_{r-1}).
\tag{12}
]

Remark 1. It can be shown that under certain modifications of formula (12), formula (11) will also change.* For example, if in (12) one takes (f(x_r,t_r)) instead of (f(x_{r-1},t_{r-1})), then in (11) it is necessary to discard the last

* See, in this connection, the remark by R. Feynman ((8), p. 185).

term in the exponent; and if one takes (\frac12[f(x_r,t_r)-f(x_{r-1},t_{r-1})]), then one must discard half of this term.

Remark 2. In the scalar case (U'_k=0) (this is also true for constant matrices (a_k)). Formula (11) in this case was obtained by a heuristic argument in (9). However, the method used there leads to an inaccuracy connected with the fact that the meaning of the expression (\Psi(x_q(t))), considered above, is not indicated. It turns out that, in order to obtain the formula derived in (9), one must take for (\Psi(x_q(t))) the expression indicated at the end of Remark 1; but then the continual integral in (11) cannot be regarded as a Lebesgue integral with respect to Wiener measure (see Theorem 5 below).

1. In the case when (L(\psi)) in equation (10) is a second-order elliptic operator, the fundamental solutions (S(t;x,y)) of the equation (\partial\psi/\partial t=L(\psi)), as is known (^{10}), are nonnegative, and therefore the set function (\mu_S(q,R)) generates a measure (\mu(x(t))) in the space (M(x_0,x_l)).

Theorem 5. Consider equation (10) in the scalar case. If in the representation (9) of its fundamental solution the relation (\lim_q \Phi(x_q(t))=\Phi(x(t))) holds, where the limit is understood in the mean square with respect to the measure (\mu(x(t))), then the integral in this representation may be regarded as a Lebesgue integral with respect to the measure (\mu(x(t))).

Proof. The theorem is applicable in the case when the operator (L(\psi)) is the generator of a Markov process for which stochastic integrals exist (see (^{11}), p. 392). In particular, this holds when (L(\psi)=a\Delta\psi), and therefore in formula (11) in the scalar case the asterisk over the integral may be omitted.

For the case (\nu=1), the representability in the form of a Lebesgue integral of the fundamental solution of equation (10) (scalar) was shown by other methods in (^{12}). In the case (\nu>1), similar results follow from (^{13}).

1. By a well-known method due to R. Feynman (^{8}), one can show the representability, in the form of a limit of continual integrals, of the fundamental solution of an equation of the form (\partial\psi/\partial t=iL(\psi)+L_1(\psi)). This device consists in considering the equation
   [
   \partial\psi/\partial t=(i+\varepsilon)L(\psi)+L_1(\psi),
   ]
   to which the results set forth above are applicable (for more detail see (^{1})), and in subsequently passing to the limit as (\varepsilon\to0). In this way, for example, one can obtain a representation of the fundamental solutions of an equation of Pauli type in a form analogous to (11).

We note that the set function (\mu(q,R)) generated by the operator ((i+\varepsilon)L(\psi)) is complex, and therefore Theorem 1 is not applicable to it. However, in this case as well, under certain conditions one can prove an analogous theorem. In particular, one can show that the function (\mu_S(q,R)), constructed from the fundamental solution
[
S(t;x,y)=(4\pi Dt)^{1/2}e^{-x^2/4Dt}
]
of the equation (\partial\psi/\partial t=D\partial^2\psi/\partial x^2) for (\operatorname{Im}D\ne0), is not of bounded variation. In this connection, the assertion made in (^{9}) (p. 93) that this equation generates a measure just as good as the Wiener measure seems to us unjustified.

Kyiv Polytechnic Institute

Received
14 IX 1960

References

1. Yu. L. Daletskii, DAN, 134, No. 5 (1960).
2. S. V. Fomin, Scientific Reports of Higher Schools, Phys.-Math. Sciences, 2 (1959).
3. E. Hille, Functional Analysis and Semigroups, IL, 1951.
4. T. Kato, J. Math. Soc. Japan, 5, 208 (1953).
5. M. A. Krasnosel’skii, S. G. Krein, P. E. Sobolevskii, DAN, 112, No. 6, 990 (1957).
6. S. G. Krein, P. E. Sobolevskii, DAN, 118, No. 2, 233 (1958).
7. G. I. Kats, DAN, 119, No. 1, 19 (1958).
8. R. Feynman, in: Problems of Causality in Quantum Mechanics, IL, 1955.
9. I. M. Gel’fand, A. M. Yaglom, Uspekhi Mat. Nauk, 11, No. 1, 77 (1956).
10. L. N. Slobodetskii, Matematicheskii Sbornik, 46, 2, 229 (1958).
11. J. L. Doob, Stochastic Processes, IL, 1956.
12. Yu. V. Prokhorov, Theory of Probability and Its Applications, 1, 177 (1956).
13. A. V. Skorokhod, Theory of Probability and Its Applications, 5, 1, 45 (1960).