MATHEMATICS

M. G. SHUR

EXCESSIVE FUNCTIONS AND ADDITIVE FUNCTIONALS OF MARKOV PROCESSES

(Presented by Academician A. N. Kolmogorov on 23 X 1961)

The concept of an excessive function associated with a Markov process was first introduced by Hunt \((^{4})\). In the study of excessive functions it was found that there is a one-to-one correspondence between a broad class of excessive functions and a certain subclass of the class of additive functionals \((^{1,5,7})\). In the present note this correspondence is subjected to further analysis.

1. Let us recall some definitions. Let \(X=(x_t,\zeta,\mathcal M_t,P_x,\theta_t)\) be a homogeneous Markov process given in a measurable topological space \((E,\mathcal E,\mathcal B)\).

We use the terminology and notation of E. B. Dynkin \((^{2})\). By \(\mathcal N_t\) we denote the \(\sigma\)-algebra generated by the events \(\{x_s(\omega)\in\Gamma,\ \zeta>s\}\), where \(s\le t,\ \Gamma\in\mathcal B\). The smallest \(\sigma\)-algebra containing all \(\mathcal N_t\) is denoted by \(\mathcal N^0\). By \(\mathcal N_{t+0}\) is denoted the \(\sigma\)-algebra of events \(A\) such that, for every \(u>t\), \(\{A,\zeta>u\}\in\mathcal N_u\). Let \(\mathfrak N\) be some \(\sigma\)-algebra \((\mathfrak N\subseteq\mathcal N^0)\). We put \(A\in\mathfrak N\) if for any finite measure \(\mu\) on \(\mathcal B\) there exist sets \(A_1,A_2\in\mathfrak N\) such that \(A_1\subseteq A\subseteq A_2\) and
\[ \int P_x\{A_2\setminus A_1\}\,\mu(dx)=0. \]

By \(\tau_U\) we denote the time of first hitting the set \(U\). More precisely, \(\tau_U=\zeta\) if \(x_t\notin U\) for all \(t>0\), and
\[ \tau_U=\inf\{t:\ t>0,\ x_t\in U\} \]
otherwise. If \(P_x\{\tau_U=0\}=1\), then the point \(x\) is called regular for \(U\).

A strictly Markov, right-continuous homogeneous process is called standard \((^{3})\) if: a) the space \((E,\mathcal E)\) is Hausdorff, locally compact, and has a countable base, and the \(\sigma\)-algebra \(\mathcal B\) consists of its Borel sets; b) if \(\tau_n\) is a nonincreasing sequence of random variables independent of the future, and \(\tau=\lim_{n\to\infty}\tau_n\), then for every \(x\in E\)
\[ x_{\tau_n}\to x_\tau \]
(\(P_x\)-almost surely on the set \(\{\tau<\zeta\}\)); c) \(\overline{\mathcal N}_{t+0}\subseteq\mathcal M_t\).

An almost Borel-measurable function \(f(x)\) \((0\le f(x)\le\infty)\) is called an excessive* function associated with \(X\), if for all \(t\ge0\) and \(x\in E\)
\[ f(x)\ge M_x f(x_t),\qquad f(x)=\lim_{s\to0}M_x f(x_s). \tag{1} \]

Let \(E_n\) be a sequence of compact sets whose union is \(E\), and such that \(E_n\) is contained in the open kernel of the compact set \(E_{n+1}\). Denote by \(\zeta_n\) the time of first hitting the set \(E\setminus E_n\). An excessive function \(f(x)\) is called harmonic if \(f(x)=M_x f(x_{\zeta_n})\). This definition does not depend on the choice of \(\{E_n\}\). We shall say that the harmonic function \(f\) is regular if
\[ f(x)=M_x\{\lim_{t\uparrow\zeta} f(x_t)\}. \]

* This means \((^{4})\) that for any finite measure \(\mu\) on \(\mathcal B\) one can specify two Borel-measurable functions \(g(x)\) and \(h(x)\) such that \(g\le f\le h\), and the probability
\[ P_x\{\text{there exists }t\ge0\text{ for which }g(x_t)\ne h(x_t)\} \]
is equal to zero \(\mu\)-almost everywhere.

We shall call a function \(\varphi_t(\omega)\) \((0 \leqslant t < \zeta(\omega),\ 0 \leqslant \varphi_t(\omega) \leqslant \infty)\) a homogeneous additive functional (or simply a functional) of the process \(X\) (1), if: 1) the function \(\varphi_t\) is \(\mathscr N_{t+0}\)-measurable; 2) \(\varphi_s+\theta_s\varphi_t=\varphi_{s+t}\), \(P_x\)-almost surely on the set \(\{\omega:s+t<\zeta(\omega)\}\), for all \(s,t\geqslant0\), \(x\in E\). Unless otherwise stated, we shall consider exclusively functionals \(\varphi_t\) that are continuous from the right, i.e., such functionals for which the function \(\varphi_t\) is right-continuous in \(t\), \(P_x\)-almost surely for all \(x\in E\). We shall say that the functional \(\varphi_t\) belongs to the class \(U\) \((^5)\), if for every finite measure \(\mu\) on \(\mathscr B\) one can indicate in the \(\sigma\)-algebra \(\mathscr N^0\) two sets \(A_1\) and \(A_2\) such that \(A_1\subseteq\{\omega:\varphi_t\) and \(x_t\), as functions of \(t\), have no common discontinuity points\(\}\subseteq A_2\) and
\[ \int_E P_x\{A_2\setminus A_1\}\mu(dx)=0. \]

We shall extend the functionals \(\varphi_t\) to \(0\leqslant t\leqslant\infty\) by setting \(\varphi_t(\omega)=\varphi_{\zeta-0}(\omega)\) for \(\zeta\leqslant t\leqslant\infty\) (the quantity \(\varphi_{\zeta-0}\) is defined \(P_x\)-almost surely). The function \(M_x\varphi_\infty\) is excessive; it is called the potential of the functional \(\varphi_t\).

An excessive function \(f(x)\) will be called singular if, for every point \(x\) at which \(f(x)\) assumes a finite value: a) there exists an increasing sequence of open sets \(G_n\) such that \(P_x\{\tau_{G_n}\uparrow \zeta\}=1\) and \(M_x f(x(\tau_{G_n}))=f(x)\); b) \(\lim_{t\uparrow \zeta} f(x_t)=0\). If \(f(x)\) is singular and \(f(x)<\infty\), then there exists neither a regular harmonic function \(g(x)\) nor a functional \(\varphi_t\) such that \(g(x)\ne0\), \(M_x\varphi_\infty\ne0\), and \(g(x)\leqslant f(x)\), \(M_x\varphi_\infty\leqslant f(x)\).

Theorem 1. Let \(X\) be a standard process. Then every excessive function \(f(x)\) taking finite values* is representable in the form
\[ f(x)=f_1(x)+f_2(x)+f_3(x), \tag{2} \]
where \(f_1(x)\) is a regular harmonic function, \(f_2(x)\) is a singular function, and \(f_3(x)\) is the potential of some functional \(\varphi_t\) from the class \(U\).

Remark 1. From the results of Meyer (see \((^6)\), Part II, § 4) it is not difficult to derive that in the decomposition (2) the functions \(f_1,f_2\), and \(f_3\) are chosen uniquely, and that if \(f_3(x)=M_x\varphi_\infty=M_x\psi_\infty\), where \(\varphi_t,\psi_t\in U\), then \(P_x\{\varphi_t=\psi_t\}=1\) for all \(x\in E\) and \(t\geqslant0\).

Proof. It is readily verified that the function
\[ F_1(x)=M_x\{\lim_{n\to\infty} f(x(\zeta_n))\} \]
is harmonic \((^6)\), Part II). If the function \(f(x)\) is bounded, then the representation (2) follows from Lemma 3 of \((^6)\) and from Theorem 1 of \((^7)\); moreover \(f_1(x)=F_1(x)\), and \(f_3(x)=0\). Let \(f(x)\) be unbounded. The functions \(h_n(x)=\min(n,f(x)-F_1(x))\) are bounded and excessive and, according to what has been said, are potentials of certain functionals \(\varphi_t^{(n)}\in U\). Let \(G_n\) be some open neighborhood of the set \(A_n=\{x:f(x)>n\}\), \(\tau_n=\tau_{G_n}\), and let \(G'_n\) be the set of points regular for \(G_n\). Then, for \(m\leqslant n\),
\[ h_n(x)=M_x\int_0^{\tau_m-0} d\varphi_t^{(n)} +M_x\int_{\tau_m+0}^{\zeta} d\varphi_t^{(n)}, \tag{3} \]
where the first of the summands is the potential of the functional
\[ \bar\varphi_t= \begin{cases} \varphi_t^{(n)}, & \text{for } 0\leqslant t<\tau_m,\\ \varphi_{\tau_m-0}^{(n)}, & \text{for } \tau_m\leqslant t<\infty, \end{cases} \]

* It can be shown that the condition that the function \(f(x)\) be finite is not essential.

for the part \(X_m\) of the process \(X\) on the set \(E\setminus G'_n\), while the second term is a harmonic function for \(X_m\). From Remark 1 it follows that, for \(n \geqslant m\) and any \(t \geqslant 0\), the functions \(\varphi_t^{(n)}(\omega)\) \(P_x\)-almost surely coincide on the set \(\{\omega:t<\tau_n\}\).

Using the equality \(\lim_{n\to\infty} P_x\{\tau_{A_n}=\zeta\}=1\), choose, for a fixed \(x\), a sequence \(G_n\) such that \(P_x\{\tau_n \uparrow \zeta\}=1\), and define \(\varphi_t\) as \(\lim_{n\to\infty}\varphi_t^{(n)}\) (\(P_x\)-almost surely). It is clear that \(\varphi_t\in U\).

Put \(m=n-1\) in equality (3), and denote the second term on the right-hand side of this equality by \(H_n(x)\). Letting now \(n\) tend to infinity in (3), we conclude that there exists the limit \(F_2(x)=\lim_{n\to\infty}H_n(x)\), and
\[ F_2(x)=f(x)-F_1(x)-M_x\varphi_\infty. \]
From the last relation it is seen that
\[ F_2(x)=\lim_{t\to0}F_2(x_t) \]
(\(P_x\)-almost surely). Moreover,
\[ f(x)-F_1(x)\geqslant H_n(x)\geqslant \int_{t<\tau_k} H_n(x_t)\,P_x\{d\omega\}, \]
if \(n\geqslant k+1\). Hence it is easy to derive the inequality
\[ F_2(x)\geqslant M_xF_2(x_t) \]
and thereby prove the excessivity of the function \(F_2(x)\).

From the harmonicity of \(H_n(x)\) for \(X_m\) (\(m<n\)) and the inequality \(f(x)-F_1(x)\geqslant H_n(x)\) it follows that \(F_2(x)\) is singular.

Thus, the functions \(f_1(x)=F_1(x)\), \(f_2(x)=F_2(x)\), and \(f_3(x)=M_x\varphi_\infty\) satisfy the condition of the theorem.

1. Consider Brownian motion \(Z_m\) in \(m\)-dimensional Euclidean space \(R_m\) (\(m\geqslant3\)). By \(|x-y|\) we shall denote the Euclidean distance between the points \(x\) and \(y\). In this case the class of excessive functions for \(Z_m\) coincides with the class of nonnegative functions superharmonic in the whole space \(R_m\), completed by the function identically equal to \(+\infty\), while the class of harmonic functions for \(Z_m\) consists of the nonnegative constants.

According to the well-known Riesz theorem \((^8)\), every nonnegative function \(f(x)\) superharmonic in the whole \(R_m\) admits the representation
\[ f(x)=C+\int_{R_m}|x-y|^{-m+2}\,\mu(dy), \]
where \(C\) is a nonnegative constant, and \(\mu\) is a Borel measure; moreover, \(C\) and \(\mu\) are uniquely determined by the function \(f(x)\). Hence one can infer that, for \(f(x)\) to be singular, it is necessary and sufficient that \(C=0\) and that the measure \(\mu\) be concentrated on some set of capacity zero. If, however, \(C=0\) and the measure \(\mu\) assigns zero mass to every set of capacity zero, then \(f(x)\) is the potential of some functional \(\varphi_t^*\).

Indeed, let \(\mu_n\) be the restriction of the measure \(\mu\) to the set \(\{x:n<f(x)\leqslant n+1\}\). Then \(\mu=\sum \mu_n\), and the functions
\[ g_n(x)=\int |x-y|^{-m+2}\,\mu_n(dy) \]
are excessive and do not exceed \(n+1\). From Theorem 1 there follows the existence of functionals \(\varphi_t^{(n)}\) whose potentials are the functions \(g_n(x)\). The expression
\[ \varphi(t)=\lim_{s\to t+}\sum_{n=0}^{\infty}\varphi_s^{(n)} \]
is the required functional (cf. \((^6)\), p. 187).

From what was said in the preceding paragraph there follows the validity of the assertion of Theorem 1 also for functions \(f(x)\), excessive for \(Z_m\), which assume

* This result was first obtained by A. D. Ventcel’ \((^9)\).

the value \(+\infty\). In this case one should set

\[ f_1(x)=C,\qquad f_2(x)=\int_A |x-y|^{-m+2}\,\mu(dy), \]

\[ f_3(x)=\int_{R_m\setminus A} |x-y|^{-m+2}\,\mu(dy), \]

where \(C\) and \(\mu\) have the same meaning as before, and \(A=\{x: f(x)=+\infty\}\).

Moscow State University
named after M. V. Lomonosov

Received
18 X 1961

REFERENCES

1. V. A. Volkonskii, Tr. Mosk. matem. obshch., 9, 143 (1960).
2. E. B. Dynkin, Foundations of the Theory of Markov Processes, Moscow, 1959.
3. E. B. Dynkin, Theory of Probability and Its Applications, 5, No. 4, 441 (1960).
4. G. A. Hunt, Illinois J. Math., 1, No. 1, No. 3 (1957); 2, No. 2 (1958).
5. P. A. Meyer, C. R., 252, 2279 (1960).
6. P. A. Meyer, Ann. Inst. Fourier, 12, 125 (1962).
7. M. G. Shur, DAN, 137, No. 4, 800 (1961).
8. M. Brelot, Éléments de la théorie classique du potentiel, Paris, 1959.
9. A. D. Wentzel, DAN, 137, No. 1, 17 (1961).