Reports of the Academy of Sciences of the USSR
1966. Volume 171, No. 1

UDC 513.88+517.948+519.217

MATHEMATICS

Yu. L. Daletskii

ELLIPTIC OPERATORS IN FUNCTIONAL DERIVATIVES AND THE DIFFUSION EQUATIONS ASSOCIATED WITH THEM

(Presented by Academician N. N. Bogolyubov on 21 XII 1965)

1°. To a second-order differential expression

\[ l_2(F)=\sum_{j,k} b_{jk}\frac{\partial^2 F}{\partial x_j\partial x_k} \]

with respect to a function \(F(x_1,\ldots,x_n)\) one may give the form

\[ l_2(F)=\operatorname{Sp}(BF''), \tag{1} \]

where \(B=\|b_{jk}\|\) is the matrix of coefficients; \(F''=\|f_{jk}\|\) is the matrix of second derivatives \(f_{jk}=\partial^2F/\partial x_j\partial x_k\). Formula (1) can be generalized to the case when \(F(x)\) is a functional on an infinite-dimensional space. For the case when the operator \(B\) is bounded, this was done in \((^1)\). Here we describe a construction that makes it possible to consider also unbounded operators \(B\).

Let us agree on some notation. Let \(\mathfrak{B}\) be a Banach space; \(\mathfrak{B}^*\) the space conjugate to it. The value of a functional \(\xi\in\mathfrak{B}^*\) on an element \(\varphi\in\mathfrak{B}\) will be denoted by the symbol \((\varphi,\xi)\). A nonlinear functional \(F(x)\) in \(\mathfrak{B}\) will be called twice continuously differentiable (class \(C_2(\mathfrak{B})\)) if the representation

\[ F(x+h)-F(x)=(h,F'(x))+(h,F''(x)h)+o(\|h\|), \tag{2} \]

holds, where \(F(x)\in\mathfrak{B}^*\), \(F''(x)\in\{\mathfrak{B}\to\mathfrak{B}^*\}\), and these expressions are continuous functions of the argument \(x\) in the corresponding norms. By \(\{\mathfrak{B}_1\to\mathfrak{B}_2\}\) we shall always mean the space of linear bounded operators acting from \(\mathfrak{B}_1\) into \(\mathfrak{B}_2\).

An operator \(A\) in a Hilbert space \(\mathfrak{H}\) will be called an operator of class \(\mathfrak{S}_1\) if it has finite absolute trace, i.e., the series

\[ \sum_{k=1}^{\infty}(A\varphi_k,\varphi_k) \]

(\(\varphi_k\) is a complete orthonormal system in \(\mathfrak{H}\)) converges absolutely, and an operator of class \(\mathfrak{S}_2\) (a Hilbert–Schmidt operator) if

\[ \sum_{k=1}^{\infty}\|A\varphi_k\|^2<\infty. \]

The symbols \(D_A, R_A\) will denote, respectively, the domain of definition and the range of the operator \(A\).

2°. Let \(\mathfrak{H}_k\) \((k=1,2)\) be Hilbert spaces, and let \(D_k\) be a linear set dense in \(\mathfrak{H}_k\). Consider a linear operator \(A\) mapping \(D_2\) into \(D_1\), and suppose that the following conditions are satisfied:

a) in \(D_k\) a new norm \(\|x\|_k^+\) can be introduced in such a way that the operator \(A\) becomes bounded:

\[ \|Ax\|_1^+\le C\|x\|_2^+ \quad (x\in D_2); \]

b) the relation

\[ \|x\|_k^+=\|T_kx\|_k \quad (x\in D_k) \]

holds, where \(T_k\) is a positive definite operator in \(\mathfrak{H}_k\), with \(D_{T_k}\supset D_k\), and there exists an inverse operator \(T_k^{-1}\) (possibly unbounded);

c) the operator \(T_1^{-1}\) belongs to the class \(\mathfrak{S}_2\).

Let \(\mathfrak H_k^+\) be the completion of \(D_k\) in the norm \(\|\cdot\|_k^+\); let \(\mathfrak H_k^-\) be the completion of \(DT_k^{-1}\) in the norm \(\|x\|_k^-=\|T_k^{-1}x\|_k\). Denote by \(\hat T_k,\hat T_k^{-1}\) the closures of the operators \(T_k,T_k^{-1}\) in the corresponding norms. Then \(\hat T_k\mathfrak H_k^+\subset \mathfrak H_k\), \(\hat T_k^{-1}\mathfrak H_k^{-1}\subset \mathfrak H_k\), and the spaces \(\mathfrak H_k^+,\mathfrak H_k^-\) can be interpreted as mutually dual, with
\[ (\varphi,\xi)=(\hat T_k\varphi,\hat T_k^{-1}\xi)_k\qquad (\varphi\in\mathfrak H_k^+,\ \xi\in\mathfrak H_k^-). \]
From condition c) there follows the embedding \(\mathfrak H_1^+\subset\mathfrak H_1\subset\mathfrak H_1^-\) and membership in the class \(\mathfrak S_1\) of every operator \(B\in\{\mathfrak H_1^-\to\mathfrak H_1^+\}\). Condition a) implies the possibility of extending the operator \(A\) to all of \(\mathfrak H_2^+\) so that \(A\in\{\mathfrak H_2^+\to\mathfrak H_1^+\}\). In this case \(A^*\in\{\mathfrak H_1^-\to\mathfrak H_2^-\}\).

Consider a nonlinear functional \(F(x)\) of class \(C_2(\mathfrak H_2^-)\). For it \(F''(x)\in\{\mathfrak H_2^-\to\mathfrak H_2^+\}\), and consequently \(AF''(x)A\in\{\mathfrak H_1^-\to\mathfrak H_1^+\}\subset\mathfrak S_1\). Thus one can introduce the second-order differential operator \(l_2(F)=\operatorname{Sp}\{AF''(x)A^*\}\), which it is natural to call elliptic.

\(3^\circ\). Consider a diffusion equation of the form
\[ \partial F/\partial t+\frac12\operatorname{Sp}\{A(x,t)F''A^*(x,t)\}+(F',a(x,t))=0, \tag{3} \]
where the coefficients \(a(x,t)\in\mathfrak H_2^-\), \(A(x,t)\in\{\mathfrak H_2^+\to\mathfrak H_2^-\}\) are sufficiently smooth. For this equation one poses the Cauchy problem on the interval \(t_0\le t\le t_1\) with condition
\[ F(x,t_1)=\Phi(x), \tag{4} \]
where \(\Phi(x)\) is a bounded functional in \(\mathfrak H_2^-\). (In connection with the consideration of the backward diffusion equation, the initial condition is imposed at the right-hand end of the interval; by the substitution \(t=-t'\) one easily passes to the usual formulation of the problem.)

The solution of problem (3)—(4), under the conditions described below, can be represented in the form of the integral
\[ F(x,t)=\int_{\mathfrak H_2^-}\Phi(y)\,\mu(t,x;t_1;dy) \tag{5} \]
with respect to the probability measure \(\mu(t,x;\tau,\cdot)\), defined on a certain \(\sigma\)-ring of subsets of the space \(\mathfrak H_2^-\) containing all Borel cylinder sets. This measure depends on the parameters \(t,\tau\in[t_0,t_1]\), \(x\in\mathfrak H_2^-\), and is the probability measure generated by a random variable \(\xi(\tau)\) taking values in the space \(\mathfrak H_2^-\) and satisfying the stochastic differential equation
\[ d\xi(\tau)=a(\xi,\tau)d\tau+A^*(\xi,\tau)dw(\tau),\qquad t\le\tau, \tag{6} \]
and the condition \(\xi(\tau)\big|_{\tau=t}=x\). Here \(w(\tau)\) is a Wiener random process with values in \(\mathfrak H_1^-\), i.e. a normal homogeneous random process with independent increments, for which the increment \(w(\tau_2)-w(\tau_1)\) has zero mean and correlation operator \((\tau_2-\tau_1)I\). The existence of such a process \(w\) follows from a known result of Minlos (see \((^2,^3)\)).

Theorem. Let the coefficients \(a(\xi,\tau)\), \(A(\xi,\tau)\) satisfy, in the corresponding norms, the Lipschitz condition in \(\xi\) with a constant independent of \(\tau\in(t_0,t_1)\), and be continuous in \(\tau\). There exists, unique up to stochastic equivalence, a solution of equation (6) satisfying the condition \(\xi(t)=x\) and continuous with probability one. This solution is a Markov process with values in \(\mathfrak H_2^-\).

If the functions \(a(\xi,\tau)\), \(A(\xi,\tau)\), \(\Phi(\xi)\) belong to the class \(C_2\) in the corresponding spaces, then formula (5) gives the solution of problem (3)—(4).

The proof of the theorem is carried out in the same way as in the finite-dimensional case (see, for example, \((^4)\)). In doing so one uses estimates of stochastic-

integrals in the space $\mathfrak{H}_1^-$, given in (1), and the Itô differential formula, which for a functional $u(t,x)$ on $\mathfrak{H}_2^-$ takes the form

\[ du(t,\xi(t))=\{u_t'(t,\xi(t))+(u_x'(t,\xi(t)),\ a)+{}^1/2\operatorname{Sp}(Au_{xx}''(t,\xi(t))A^*)\}\,dt+ \]
\[ +\,(u_x'(t,\xi(t)),A^*dw). \tag{7} \]

$4^\circ$. Let us consider several special cases.

1) Let $A(t)\in \mathfrak{S}_2(t_0\le t\le t_1)$ and suppose there exists $\tau\in[t_0,t_1]$ for which the operator $A^{-1}(\tau)$ exists (unbounded), with $R_{A(t)}\subseteq R_{A(\tau)}$. Put $T_1=|A(\tau)|^{-1}$, $T_2=I$. Then $\mathfrak{H}_1^+=\bar R_{A(\tau)}\subset \mathfrak{H}$; $\mathfrak{H}_2^+=\mathfrak{H}_2^-=\mathfrak{H}$, and thus the measure $\mu$ turns out to be concentrated in $\mathfrak{H}$. The operator $B=A^*A\in\mathfrak{S}_1$ has meaning in $\mathfrak{H}$. If one takes $\mathfrak{H}=\mathcal L_2[a,b]$, then to this operator there corresponds a certain kernel $b(s_1,s_2)$; let $\delta^2F/\delta x(s_1)\delta x(s_2)$ be the generalized kernel corresponding to the operator $F''$ (the second variational derivative of the functional $F(x)$). The operator $l_2(F)$ takes the form

\[ l_2=\int_a^b\int_a^b b(s_1,s_2) \frac{\delta^2F}{\delta x(s_1)\delta x(s_2)}\,ds_1ds_2. \]

Equations with such operators were in fact considered in $(^5,^6)$. We note that the general stochastic equation (6) can be reduced to an equation with an operator $A\in\mathfrak{S}_2$ by expanding the process $w(t)$ in the eigenvectors of the operator $T$ (see $(^1)$).

The case when $A^m\in\mathfrak{S}_2$ is considered analogously. Imposing on the operator $A(t)$ more stringent requirements (for example, $A=A_1^k$, where $A_1\in\mathfrak{S}_2$), we shall obtain equations of the form (3), for which the Cauchy problem is solvable for broader classes of functionals $\Phi(x)$ (for example, those depending on derivatives of the function $x(s)$).

2) The operator $A(t)$ is bounded in $\mathfrak{H}$ and leaves invariant some domain $D_T$, where $T^{-1}\in\mathfrak{S}_2$. This case was considered in $(^1)$. If it is known only that $A(t)$ is bounded in $\mathfrak{H}$, then one can enlarge $\mathfrak{H}$ by completing it with respect to a certain weaker norm of the form $\|x\|_-=\|T^{-1}x\|$ and take $\mathfrak{H}_1^+=\mathfrak{H}_2^+=\mathfrak{H}$, taking the completed space as the basic one. In this case the measure will be concentrated in an even broader space.

3) Let, generally speaking, $A(t)$ be an unbounded operator in $\mathfrak{H}$ having the property that, for some $\tau\in[t_0,t_1]$, $A^{-1}(\tau)\in\mathfrak{S}_2$, and moreover
$D_{A(t)A(\tau)}\supseteq D_{A^2(\tau)}$ (this is the case, for example, if the domain $D_{A^2(t)}$ is constant except at isolated points where $A(t)=0$). In this case one may take $T_1=|A(\tau)|$, $T_2=|A^2(\tau)|$, $\mathfrak{H}_1^+=D_{A(\tau)}$, $\mathfrak{H}_2^+=D_{A^2(\tau)}$. Cases in which the condition $A^{-\alpha}(\tau)\in\mathfrak{S}_2$ holds for some $\alpha$ greater or less than one are considered analogously. In the case of the space $\mathfrak{H}=\mathcal L_2[a,b]$, the considerations set forth make it possible to consider equations (3) and (6) with a differential operator $A(t)$.

$5^\circ$. Consider a sequence of equations of type (6), whose coefficients $a_n(\xi,t)$ and $A_n(\xi,t)$ $(n=0,1,2,\ldots)$ satisfy the conditions of Theorem 1, and let $\xi_n(t)$ be solutions of these equations satisfying one and the same condition $\xi_n(t_0)=x$.

Theorem 2. Let, for every $x\in\mathfrak{H}_1^-$,

\[ \lim_{n\to\infty}\|A_0^*(\xi,\tau)x-A_n^*(\xi,\tau)x\|_2^-=0 \quad\text{and}\quad \lim_{n\to\infty}\|a_n(\xi,\tau)-a_0(\xi,\tau)\|_2^-=0 \quad(\xi\in\mathfrak{H}_{-2}, \]

$t_0\le \tau\le t_1$), and suppose that the functions $a_n(\xi,\tau)$ and $A_n(\xi,\tau)$ have one and the same Lipschitz constant. Then

\[ \lim_{n\to\infty}\left\{\sup_t M\|\xi_0(t)-\xi_n(t)\|^2\right\}=0. \]

In this case

\[ F_0(x,t)=\lim_{n\to\infty} F_n(x,t), \]

where \(F_n\) is the solution of the Cauchy problem (5), (3)—(4) with coefficients \(a_n, A_n\) \((n=0,1,\ldots)\).

Using this theorem, by approximating, in the strong sense, the coefficients of problem (3)—(4) by sequences of finite-dimensional operators, we can represent the solution of this problem as the limit of finite-dimensional problems. Since the solutions of these problems are unique in the class of functionals under consideration, in this way one can obtain a proof of the uniqueness theorem for the solution of problem (3)—(4).

Kyiv Polytechnic
Institute

Received
16 XII 1965

REFERENCES

1. Yu. L. Daletskii, DAN, 166, No. 5 (1966).
2. R. A. Minlos, Tr. Mosk. matem. obshch., 8, 497 (1959).
3. I. M. Gel'fand, N. Ya. Vilenkin, Generalized Functions, vol. 4, Some Applications of Harmonic Analysis, Moscow, 1961.
4. I. I. Gikhman, A. V. Skorokhod, Introduction to the Theory of Random Processes, Moscow, 1965.
5. V. V. Baklan, Dokl. AN USSR, No. 5, 554 (1965).
6. T. L. Chantladze, Soobshch. AN GruzSSR, 23, 3, 527 (1964).