Reports of the Academy of Sciences of the USSR
1960. Volume 132, No. 6

MATHEMATICS

V. Yu. Krylov

ON SOME PROPERTIES OF THE DISTRIBUTION CORRESPONDING TO THE EQUATION

\[ \frac{\partial u}{\partial t}=(-1)^{q+1}\frac{\partial^{2q}u}{\partial x^{2q}} \]

(Presented by Academician A. N. Kolmogorov on 27 II 1960)

As early as the 1920s, N. Wiener \((^{1,2})\) considered a probability distribution in the space of continuous functions—the trajectories of Brownian particles—associated with the heat equation. This distribution has since been studied by many authors (see the bibliography in \((^{3})\)) and is usually called Wiener measure. I. M. Gelfand posed the problem of studying distributions analogous to Wiener measure that are associated with differential equations of a more general form \((^{3},\ p. 87)\).

The present work is devoted to the study of certain properties of a distribution in the space \(C[0,T]\) of functions \(x(t)\), continuous on \([0,T]\), corresponding to the equation

\[ \frac{\partial u}{\partial t}=(-1)^{q+1}\frac{\partial^{2q}u}{\partial x^{2q}} . \tag{1} \]

This distribution is defined, as usual, on cylindrical subsets (quasi-intervals) of the space \(C[0,T]\). It is proved here that the mean, with respect to the distribution, of the functional
\[ \exp\left\{-\int_0^T V[x(t)]\,dt\right\} \]
exists and is a solution of the equation

\[ \frac{\partial u}{\partial t}=(-1)^{q+1}\frac{\partial^{2q}u}{\partial x^{2q}}-V(x)u \tag{2} \]

for a certain class of functions \(V(x)\). Thus the existence of the distribution, or generalized measure \(P_{2q}\), is proved already in the whole space \(C[0,T]\) of continuous functions. It turns out that the total variation on \(C[0,T]\) of the measure \(P_{2q}\) for \(q>1\) is infinite. In the present work it is established that the measure \(P_{2q}\) is concentrated on a compact set in the sense that its total variation outside this compact set can be made arbitrarily small. In addition, for any \(q>1\) the “arcsine law” known for Wiener measure is generalized \((^{3},\ p. 90)\).

A measure in the space \(C[0,T]\) of functions \(x(t)\), continuous on the interval \([0,T]\) and satisfying the condition \(x(0)=0\), is defined as follows. Divide the interval \([0,T]\) into \(n\) equal parts, each of length \(\Delta=T/n\), by the points \(t_k=kT/n\) \((k=0,\ldots,n)\).

Let the set \(I_n\) of functions \(x(t)\in C[0,T]\) satisfying the inequalities \(a_k\leq x(t_k)\leq b_k\) \((k=1,\ldots,n)\) be called a quasi-interval. Define the measure \(P_{2q}\{I_n\}\) of the quasi-interval \(I_n\) by the formula

\[ P_{2q}\{I_n\}= \int_{a_1}^{b_1}\cdots\int_{a_n}^{b_n} \prod_{k=0}^{n-1}G(t_{k+1}-t_k,\ x_{k+1}-x_k)\,dx_{k+1}, \tag{3} \]

where \(x_0=0;\ t_0=0;\ G(t,x)\) is the Green’s function

\[ G(t,x)=t^{-1/2q}g\bigl(xt^{-1/2q}\bigr), \qquad g(y)=\frac{1}{2\pi}\int_{-\infty}^{+\infty} e^{-\xi^{2q}+iy\xi}\,d\xi \]

of equation (1)*. For each integer \(m\le n\), define the function

\[ u_m(t,x)=\int_{-\infty}^{+\infty}\cdots\int \psi(x_0)\exp\left\{-\Delta\sum_{k=0}^{m-1}V(x_{k+1})\right\} \prod_{k=0}^{m-1}G(\Delta,x_{k+1}-x_k)\,dx_k, \]

where \(x_m=x(t_m)=x(t)=x,\ t_m=m\Delta\).

Theorem 1. Let \(V(x)\) be a bounded smooth function. Then the sequence \(u_m(t,x)\) converges uniformly in \(x\) on every finite interval to a limit \(u(t,x)\), which is the solution of the Cauchy problem for equation (2) with initial condition \(u(0,x)=\psi(x)\).

The assertion of Theorem 1 follows directly from the compactness of the family of functions \(u_m^\Delta(x)=u_m(t,x)\). The compactness of the family of functions \(u_m^\Delta(x)\) is easily derived from the relation

\[ u_{m+1}^\Delta(x)=e^{-\Delta V(x)}\int_{-\infty}^{+\infty}G(\Delta,x-\xi)u_m^\Delta(\xi)\,d\xi . \]

Theorem 2. The measure \(P_{2q}\{C\}\) has unbounded total variation for \(q>1\).

Proof. The measure \(P_{2q}\{C\}\) is given by a density; therefore its total variation on \(C[0,T]\) is equal to the integral over \(C[0,T]\) of the modulus of the density

\[ \underset{C}{\mathbf V}P_{2q} = \lim_{n\to\infty} \int_{-\infty}^{+\infty}\cdots\int \prod_{k=0}^{n-1}\left|G(\Delta,x_{k+1}-x_k)\right|\,dx_{k+1}. \]

Taking into account that the function \(g(y)\) for \(q>1\) is sign-changing and that

\[ \int_{-\infty}^{+\infty}g(y)\,dy=1, \]

and consequently

\[ \int_{-\infty}^{+\infty}|g(y)|\,dy>1, \]

we obtain

\[ \underset{C}{\mathbf V}P_{2q} = \lim_{n\to\infty} \left(\int_{-\infty}^{+\infty}|g(y)|\,dy\right)^n = \infty, \]

which proves Theorem 2.

Let \(t\) vary on the interval \([0,1]\). Consider the set \(A_H^p\subset C[0,1]\) of continuous functions \(x(t)\), equal to zero at \(t=0\), for each of which there exist points \(t_1\) and \(t_2\) on the interval \([0,1]\) such that

\[ |x(t_1)-x(t_2)|>H\gamma |t_1-t_2|^{1/p}, \]

where \(H\) and \(\gamma\) are some positive numbers. The set complementary to \(A_H^p\) is, obviously, compact for any \(p>0\).

Theorem 3. The total variation of the measure \(P_{2q}\{A_H^p\}\) on the set \(A_H^p\) tends to zero as \(H\to\infty\), if \(1/p=1/2q-\varepsilon\), where \(\varepsilon\) is any positive number.

The proof of this theorem will rely on the following lemma, which we merely state.

* A formula analogous to (3) for the case of the Schrödinger equation is found in the work of R. P. Feynman (4), and for equations of a more general form was given by L. V. Kobelev (unpublished).

Lemma. If the function \(x(t)\in A_H^p\), then there exist integers \(m,i\) (with \(m\leqslant 2^i\)) such that

\[ \left|x[m2^{-i}]-x[(m+1)2^{-i}]\right|>H2^{-i/p}, \tag{4} \]

for a suitable choice of \(\gamma=\gamma(p)\).

We now pass to the proof of Theorem 3. By Lemma 1, for any function from the set \(A_H^p\) there exist integers \(m\) and \(i\) such that inequality (4) is satisfied. Therefore it is enough for us to estimate the variation of the measure \(P_{2q}\) on functions satisfying (4) for all \(m\leqslant 2^i\) and all \(i\). Obviously, this variation is equal to

\[ \operatorname{V}_{A_H^p} P_{2q} = \sum_{i=0}^{\infty}2^{i+1} \int_{H2^{-i/p}}^{\infty} \left|G(2^{-i},\xi)\right|\,d\xi = \sum_{i=0}^{\infty}2^{i+1} \int_{H2^{\,i(1/2q-1/p)}}^{\infty} |g(y)|\,dy, \tag{5} \]

where \(y=\xi 2^{i/2q}\). Since \(|g(y)|<\exp\{-Cy\}\) for large \(y\) and for any \(q\geqslant 1\) (see, for example, (5), p. 120), it follows that for \(1/2q-1/p=\varepsilon>0\) the sum of the series (5) tends to zero as \(H\to\infty\), which is what was asserted in Theorem 3.

Finally, let us consider one more property of the measure \(P_{2q}\). For a Brownian particle the “arcsine law” is known. According to it, the probability \(F(t_1)\) that a particle which is at the origin at the moment \(t=0\) is on the positive half-axis during the time \(T\) for a time not less than \(t_1\) is equal to

\[ F(t_1)=\frac{2}{\pi}\arcsin\sqrt{\frac{t_1}{T}}. \tag{6} \]

An unexpected circumstance is that this formula is valid for any \(q\geqslant 1\).

Theorem 4. For any \(q\geqslant 1\), the full measure \(F(t_1)\) of the set of those trajectories \(x(t)\in C[0,T]\) which, on the interval \([0,T]\), are positive during a time not less than \(t_1\), is given by formula (6).

Proof. Obviously,

\[ \Phi(T)= \int_{C[0,T]} \exp\left\{ -\int_{\substack{x(t)>0\\ t\leqslant T}} dt \right\} dP_{2q}[x(t)] = \int_0^T e^{-t_1}\,dF(t_1). \tag{7} \]

Thus we need to find the value of the integral appearing on the left-hand side of equality (7). Let \(\widetilde u(t,x)\) be the solution of equation (2) with potential \(V(x)\), equal to 1 for \(x>0\) and 0 for \(x<0\), satisfying the conditions

\[ \lim_{x\to\pm\infty}\widetilde u(t,x)=0,\qquad \widetilde u(0,x)=\delta(x). \]

The Laplace transform \(\varphi(x)\) of the function \(\widetilde u(t,x)\),

\[ \varphi(x)=\int_0^\infty \widetilde u(t,x)e^{-Et}\,dt, \]

will satisfy the equation

\[ (-1)^{q+1}\varphi^{(2q)}(x)=(V(x)+E)\varphi(x) \]

and the conditions

\[ \lim_{x\to\pm\infty}\varphi(x)=0,\qquad \varphi(+0)=\varphi(-0),\ldots,\quad \varphi^{(2q-2)}(+0)=\varphi^{(2q-2)}(-0), \]

\[ \varphi^{(2q-1)}(+0)-\varphi^{(2q-1)}(-0)=(-1)^q. \]

By virtue of the assertion of Theorem 1, we have

\[ \int_0^\infty \Phi(T)e^{-ET}\,dT = \int_{-\infty}^{+\infty}\varphi(x)\,dx. \tag{8} \]

Introduce the notation \(a_k=\sqrt[2q]{E}\,\varepsilon_k\) for \(k=1,\ldots,q\) and \(a_k=\sqrt[2q]{E+1}\,\varepsilon_k\) for \(k=q+1,\ldots,2q\), where \(\varepsilon_k\) are the roots of the equation \(\varepsilon^{2q}=(-1)^{q+1}\), numbered so that \(\operatorname{Re}\varepsilon_k>0\) for \(k=1,2,\ldots,q\).

It is easy to verify that, in this notation, the desired function
\(\varphi(x)=W^{-1}\sum_{k=q+1}^{2q} W_k\exp(a_k x)\) for \(x>0\), and
\(\varphi(x)=-W^{-1}\sum_{k=1}^{q} W_k\exp(a_k x)\) for \(x<0\), where \(W\) is the Vandermonde determinant of order \(2q\) formed from \(a_k\) \((k=1,2,\ldots,2q)\), and \(W_k\) is obtained from \(W\) by replacing the \(k\)-th column by the vector \(\{0,\ldots,0,(-1)^q\}\). The integral appearing on the right-hand side of equality (8) is equal to

\[ \int_{-\infty}^{+\infty}\varphi(x)\,dx = -\sum_{k=1}^{q}\frac{W_k}{a_k W} = \]

\[ = \frac{(-1)^{q+1}}{W}\sum_{k=1}^{2q}\frac{(-1)^k}{a_k} \left| \begin{array}{cccccc} 1 & \ldots & 1 & 1 & \ldots & 1\\ a_1 & \ldots & a_{k-1} & a_{k+1} & \ldots & a_{2q}\\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\ a_1^{2q-2} & \ldots & a_{k-1}^{2q-2} & a_{k+1}^{2q-2} & \ldots & a_{2q}^{2q-2} \end{array} \right| = \frac{(-1)^{q+1}}{W\prod_{i=1}^{2q}a_i} \left| \begin{array}{cccc} 1 & \ldots & 1\\ a_1 & \ldots & a_{2q}\\ \cdot & \cdot & \cdot\\ a_1^{2q-2} & \ldots & a_{2q}^{2q-2}\\ \prod_{i\ne 1}^{2q}a_i & \ldots & \prod_{i\ne 2q}^{2q}a_i \end{array} \right| = \]

\[ = (-1)^q\left(\prod_{i=1}^{2q}a_i\right)^{-1} = \frac{1}{\sqrt{E(1+E)}} \]

by the definition of the quantities \(a_k\). Inverting the Laplace transform (8), we obtain

\[ \Phi(T)=\frac{1}{\pi}\int_0^T \frac{e^{-t_1}\,dt_1}{\sqrt{t_1(T-t_1)}} = \int_0^T e^{-t_1}\,dF(t_1), \]

which was required to be proved.

I take this opportunity to express my gratitude to my scientific adviser, Corresponding Member of the Academy of Sciences of the USSR I. M. Gel'fand, and to I. I. Pyatetskii-Shapiro for a number of valuable suggestions.

Received
25 II 1960

References

1. N. Wiener, J. Math. and Phys., 2, 131 (1923).
2. N. Wiener, Proc. London Math. Soc., ser. 2, 22, No. 6, 454 (1924).
3. I. M. Gel'fand, A. M. Yaglom, Russian Mathematical Surveys, 11, no. 1 (67), 77 (1956).
4. R. P. Feynman, Rev. Mod. Phys., 20, No. 2, 367 (1948).
5. M. A. Evgrafov, Asymptotic Estimates and Entire Functions, Moscow, 1957.