Full Text
Reports of the Academy of Sciences of the USSR
1965. Volume 163, No. 2
MATHEMATICS
V. Yu. KRYLOV
INTEGRATION OF ANALYTIC FUNCTIONALS WITH RESPECT TO SIGN-ALTERNATING DISTRIBUTIONS
(Presented by Academician A. N. Kolmogorov, 28 XII 1964)
The purpose of this note is to prove the integrability of a certain class of analytic functionals with respect to the sign-alternating distribution \(P_{2q}\) corresponding to the equation
\[ \partial u/\partial t = (-1)^{q+1}\partial^{2q}u/\partial x^{2q}. \tag{1} \]
The distribution \(P_{2q}\) is defined in the space of continuous functions \(x(t)\), \(x(0)=0\), and is a generalization of Wiener measure \((^1)\) associated with the heat equation.
Let \(C_0[0,1]\) denote the space of functions \(x(t)\) continuous on \([0,1]\), with \(x(0)=0\). On all quasi-intervals \(K_n(t_k,a_k,b_k)\) of the space \(C_0[0,1]\) we define a consistent system of distributions, defining the weight \(P_{2q}[K_n]\) of the quasi-interval \(K_n(t_k,a_k,b_k)\) by the formula \((^2)\)
\[ P_{2q}[K_n] = \int_{a_1}^{b_1}\cdots\int_{a_n}^{b_n} \prod_{k=0}^{n-1} G_{2q}\bigl(t_{k+1}-t_k,\; x_{k+1}-x_k\bigr)\,dx_{k+1}, \]
where
\[ G_{2q}(t,x) = \frac{1}{2\pi}\int_{-\infty}^{+\infty} \exp\bigl[-t\xi^{2q}+ix\xi\bigr]\,d\xi \]
is the Green function of equation (1).
Suppose that on the space \(C_0[0,1]\) a functional \(F[x(t)]\) is given. Denote by \(F[x_1,\ldots,x_n]\) the value of the functional \(F[x(t)]\) on the stepwise curve \(x^s(t)=x_k\) for \(t_k\le t<t_{k+1}\) \((k=0,\ldots,n-1)\), corresponding to the partition \(0<t_1<\cdots<t_n\le 1\).
We shall say that there exists a continual integral
\[ I=\int_{C_0} F[x(t)]\,dP_{2q}[x(t)] \]
of the functional \(F[x(t)]\) with respect to the distribution \(P_{2q}\) over the whole space \(C_0[0,1]\), if there exists a limit \(I=\lim_{\lambda\to 0} I_n\), where \(\lambda=\max_k(t_{k+1}-t_k)\), of the \(n\)-fold integral
\[ I_n= \int_{-\infty}^{+\infty}\cdots\int F[x_1,\ldots,x_n] \prod_{k=0}^{n-1} G_{2q}\bigl(t_{k+1}-t_k,\;x_{k+1}-x_k\bigr)\,dx_{k+1}, \]
independent of the partition of the interval \([0,1]\) by the points \(t_k\) \((k=1,\ldots,n)\).
A functional \(F[x(t)]\), defined on \(C_0[0,1]\), will be called analytic if it is representable in the form
\[ F[x(t)] = \sum_{p=0}^{\infty}\frac{1}{p!}F_p[x(t)], \tag{2} \]
where
\[ F_p[x(t)] = \int_0^1 \cdots \int_0^1 h_p(t_1,\ldots,t_p)\,dx(t_1)\cdots dx(t_p), \]
the functions \(h_p(t_1,\ldots,t_p)\) being assumed to depend symmetrically on their arguments.
Moreover, in order that the integral in the definition of \(F_p[x(t)]\) have meaning for an arbitrary continuous function \(x(t)\in C_0[0,1]\), and also for a step curve \(x^s(t)\), we require that the function \(h_p(t_1,\ldots,t_p)\) and all its derivatives up to order \(p\), taken with respect to each argument no more than once, be continuous and equal to zero if at least one variable \(t_j=1\) \((j=1,\ldots,p)\) identically with respect to the remaining variables. In this case, by definition, we set
\[ F_p[x(t)] = (-1)^p \int_0^1 \cdots \int_0^1 \frac{\partial^p h_p(t_1,\ldots,t_p)}{\partial t_1\cdots \partial t_p} x(t_1)\cdots x(t_p)\,dt_1\cdots dt_p. \]
Theorem*. The continual integral \(I\) of the analytic functional \(F[x(t)]\) with respect to the distribution \(P_{2q}\) on the space \(C_0[0,1]\) exists if, for \(p=2ql\) \((l=1,2,\ldots)\), the functions \(h_p(t_1,\ldots,t_p)\) satisfy the inequalities
\[ \left|h_{2ql}(s_1,\ldots,s_1,s_2,\ldots,s_2,\ldots,s_l,\ldots,s_l)\right|<c^l, \tag{3} \]
where the constant \(c>0\) does not depend on \(l\), and in the arguments of the function \(h_{2ql}(s_1,\ldots,s_1,s_2,\ldots,s_2,\ldots,s_l,\ldots,s_l)\) there are groups of \(2q\) identical terms, \(0\le s_i\le 1\) \((i=1,\ldots,l)\). Moreover, the equality holds
\[ I=\sum_{l=0}^{\infty}\frac{(-1)^{l(q+1)}}{l!} \int_0^1\cdots\int_0^1 h_{2ql}(s_1,\ldots,s_1,\ldots,s_l,\ldots,s_l)\,ds_1\cdots ds_l. \tag{4} \]
Proof. The integral of the functional \(F_p[x(t)]\), by definition, is equal to the limit, as \(\max_k \Delta t_{k+1}\to 0\) \((\Delta t_{k+1}=t_{k+1}-t_k)\), of the \(2n\)-fold integral
\[ I_n^p=\frac{1}{(2\pi)^n}\int\cdots\int_{-\infty}^{+\infty} F_p[x_1,\ldots,x_n]\times \]
\[ \times \exp\left[ -\sum_{k=0}^{n-1}\xi_{k+1}^{2q}\Delta t_{k+1} +i\sum_{k=0}^{n-1}(x_{k+1}-x_k)\xi_{k+1} \right]dx_1\cdots dx_n\,d\xi_1\cdots d\xi_n. \]
Here \(F_p[x_1,\ldots,x_n]\) is the value of the functional \(F_p[x^s(t)]\) on the step curve \(x^s(t)\). By the definition of the functional \(F_p[x(t)]\) it is equal to
\[ F_p[x_1,\ldots,x_n] = (-1)^p \sum_{0<k_1,\ldots,k_p\le n} \Delta^p h_p(t_{k_1},\ldots,t_{k_p})\,x_{k_1}\cdots x_{k_p} = \]
\[ = \sum_{0\le k_1,\ldots,k_p\le n-1} h_p(t_{k_1},\ldots,t_{k_p}) (x_{k_1+1}-x_{k_1})\cdots(x_{k_p+1}-x_{k_p}). \]
Let us note that \(F_p[x_1,\ldots,x_n]\) is a homogeneous polynomial of degree \(p\) in \(x_1,\ldots,x_n\). Making in \(I_n^p\) the triangular change of variables
\(x_{k+1}=z_1\Delta t_1+\cdots+z_{k+1}\Delta t_{k+1}\),
\(\xi_{k+1}\Delta t_{k+1}=\eta_{k+1}\) \((k=0,\ldots,n-1)\), and introducing the notation
\(H_p[z_1,\ldots,z_n]=F_p[z_1\Delta t_1,\ldots,z_1\Delta t_1+\cdots+z_n\Delta t_n]\), we obtain
* E. V. Maikov in [3] gave another definition of integrals with respect to generalized measures for functionals of a more general type, which he called \(\tau\)-analytic, without carrying out the computation of the integrals for concrete measures.
\[ I_n^p=\frac{1}{(2\pi)^n}\int_{-\infty}^{+\infty}\cdots\int H_p[z_1,\ldots,z_n]\times \]
\[ \times \exp\left[-\sum_{k=0}^{n-1}\frac{\eta_{k+1}^{2q}}{(\Delta t_{k+1})^{2q}}\Delta t_{k+1} +i\sum_{k=0}^{n-1}z_{k+1}\eta_{k+1}\right]\,dz_1\ldots dz_n\,d\eta_1\ldots d\eta_n . \]
Since \(H_p[z_1,\ldots,z_n]\) is a homogeneous polynomial of degree \(p\) in \(z_1,\ldots,z_n\), we have \(I_n^p=0\) if \(p\) is not divisible by \(2q\). If \(p=2ql\), then for \(I_n^{2ql}\) we obtain the expression
\[ I_n^{2ql}=\frac{(-1)^{l(q+1)}}{l!}\times \]
\[ \times \left. \sum_{l_1+\cdots+l_n=l} \frac{l!}{l_1!\cdots l_n!} \frac{\partial^{2ql}H_{2ql}[z_1,\ldots,z_n](\Delta t_1)^{l_1}\cdots(\Delta t_n)^{l_n}} {\partial z_1^{2ql_1}\cdots \partial z_n^{2ql_n}(\Delta t_1)^{2ql_1}\cdots(\Delta t_n)^{2ql_n}} \right|_{z_1=\cdots=z_n=0}. \]
By definition,
\[ H_{2ql}[z_1,\ldots,z_n] = \sum_{0\le k_1,\ldots,k_{2ql}\le n-1} h_{2ql}(t_{k_1},\ldots,t_{k_{2ql}}) z_{k_1}\Delta t_{k_1}\cdots z_{k_{2ql}}\Delta t_{k_{2ql}}, \]
where \(h_p(t_1,\ldots,t_p)\) is a symmetric function of its arguments, so that
\[ \left. \frac{\partial^{2ql}H[z_1,\ldots,z_n]} {\partial z_1^{2ql_1}\cdots \partial z_n^{2ql_n}} \right|_{z_1=\cdots=z_n=0} = (2ql)!\,h_{2ql}(t_1,\ldots,t_1,\ldots,t_n,\ldots,t_n)(\Delta t_1)^{2ql_1}\cdots(\Delta t_n)^{2ql_n}, \]
where in the arguments of the function \(h_{2ql}(t_1,\ldots,t_1,\ldots,t_n,\ldots,t_n)\) there are groups of \(2ql_1,\ldots,2ql_n\) identical \(t_1,\ldots,t_n\), respectively. Consequently, for \(I_n^{2ql}\) we obtain
\[ I_n^{2ql}\frac{l!}{(-1)^{l(q+1)}(2ql)!} = \sum_{l_1+\cdots+l_n=l} \frac{l!}{l_1!\cdots l_n!} h_{2ql}(t_1,\ldots,t_1,\ldots,t_n,\ldots,t_n)\times \]
\[ \times(\Delta t_1)^{l_1}\cdots(\Delta t_n)^{l_n} = \]
\[ = \frac{(-1)^{l(q+1)}(2ql)!}{l!} \sum_{0<j_1,\ldots,j_l\le n} h_{2ql}(t_{j_1},\ldots,t_{j_1},\ldots,t_{j_l},\ldots,t_{j_l}) \Delta t_{j_1}\cdots \Delta t_{j_l}. \tag{5} \]
The approximate value \(I_n\) of the analytic functional (2) for any fixed partition of the interval \([0,1]\) is defined as the limit
\[ I_n=\lim_{M\to\infty}\sum_{l=0}^{M}\frac{1}{(2ql)!}I_n^{2ql}. \tag{6} \]
This limit exists under the assumptions made in (3), since for any \(\varepsilon>0\) and any partition of the interval \([0,1]\) there exists a number \(M\) such that for all \(M'>M\), in view of formula (5),
\[ \left| \sum_{l=M'}^{\infty}\frac{1}{(2ql)!}I_n^{2ql} \right| \le \sum_{l=M'}^{\infty}\frac{c^l}{l!}<\frac{\varepsilon}{2}. \]
On the other hand, since the functions \(h_p(t_1,\ldots,t_p)\) are continuous by assumption, one can choose \(\delta=\delta(\varepsilon)\) such that for all partitions for which \(\lambda=\max_k \Delta t_k<\delta\), the inequality
\[ \left| \sum_{l=0}^{M}\frac{1}{(2ql)!}I_n^{2ql} - \sum_{l=0}^{M}\frac{1}{l!} \int_0^1\cdots\int_0^1 h_{2ql}(s_1,\ldots,s_1,\ldots,s_l,\ldots,s_l)\,ds_1\cdots ds_l \right| < \frac{\varepsilon}{2} \]
holds, since expression (5) is an integral sum for the integral of the continuous function \(h_{2ql}(s_1,\ldots,s_1,\ldots,s_l,\ldots,s_l)\) over the \(l\)-dimensional cube. The theorem is proved.
We note that for the integral \(I_n\) (6) the formula holds
\[ I_n=\sum_{l=0}^{\infty}\frac{(-1)^{l(q+1)}}{l!}\times \]
\[ \times \sum_{l_1+\cdots+l_n=l} \frac{l!}{l_1!\cdots l_n!}\, \frac{ \dfrac{\partial^{2ql}H_{2ql}[z_1,\ldots,z_n](\Delta t_1)^{l_1}\cdots(\Delta t_n)^{l_n}} {\partial z_1^{2ql_1}\cdots \partial z_n^{2ql_n}(\Delta t_1)^{2ql_1}\cdots(\Delta t_n)^{2ql_n}} }{} \bigg|_{z_1=\cdots=z_n=0}, \]
which is conveniently written in the form
\[ I_n=\exp\left\{(-1)^{q+1}\sum_{k=0}^{n-1} \frac{\partial^{2q}}{\partial z_{k+1}^{2q}(\Delta t_{k+1})^{2q}}\,\Delta t_{k+1}\right\} H[z_1,\ldots,z_n]\bigg|_{z_1=\cdots=z_n=0}. \]
For the analytic functional \(F[x(t)]\), by what has been proved, there exists the limit of \(I_n\) as \(\max_k \Delta t_k\to 0\). We make this formal passage to the limit in the last expression for \(I_n\). We obtain the formula
\[ I=\exp\left\{(-1)^{q+1}\int_0^1 \frac{\delta^{2q}}{[\delta z(t)]^{2q}}\,dt\right\} F\left[\int_0^t z(s)\,ds\right]\bigg|_{z(s)\equiv 0}, \tag{7} \]
which reduces continual integration of a functional to the application to it, at \(z(s)\equiv 0\), of an operator in variational derivatives. We note that expression (4) can also be obtained formally from (7), using the fact that for an analytic functional
\[ \left\{ \delta^p F\left[\int_0^t z(s)\,ds\right]/ \delta z(t_1)\cdots \delta z(t_p) \right\}\bigg|_{z(s)\equiv 0} = h_p(t_1,\ldots,t_p). \]
As an example of the application of the theorem and formula (7), let us compute the characteristic functional \(\Phi[y(t)]\) of the distribution \(P_{2q}\).
The characteristic functional \(\Phi[y(t)]\) of the distribution \(P_{2q}\) is the continual integral over the entire space \(C_0[0,1]\) of the functional
\[ E[x(t)]=\exp\left[i\int_0^1 x(t)y(t)\,dt\right]. \]
Since the functional \(E[x(t)]\) satisfies the conditions of the theorem for every continuous function \(y(t)\), \(y(1)=0\), the characteristic functional \(\Phi[y(t)]\) exists. Formula (7) in fact makes it possible to compute \(\Phi[y(t)]\). Indeed, by formula (7) we have
\[ \Phi[y(t)] = \exp\left\{(-1)^{q+1}\int_0^1 \frac{\delta^{2q}}{[\delta z(t)]^{2q}}\,dt\right\} \exp\left\{ i\int_0^1 y(t)\left[\int_0^t z(s)\,ds\right]dt \right\}\bigg|_{z(s)\equiv 0}. \]
Changing the order of integration in the last exponent and noting that
\[ \frac{\delta^p}{[\delta z(t)]^p} \exp\left\{ i\int_0^1 z(t)\left[\int_t^1 y(s)\,ds\right]dt \right\} \bigg|_{z(s)\equiv 0} = \left[i\int_t^1 y(s)\,ds\right]^p, \]
we finally obtain for \(\Phi[y(t)]\) the expression
\[ \Phi[y(t)] = \exp\left\{ -\int_0^1\left[\int_t^1 y(s)\,ds\right]^{2q}dt \right\}. \]
Received
15 XII 1964
REFERENCES
- N. Wiener, Proc. London Math. Soc., Ser. 2, 22, No. 6, 454 (1924).
- V. Yu. Krylov, DAN, 132, No. 6, 1254 (1960).
- E. V. Maikov, UMN, 18, 3, 243 (1963).