Keller kindly told us of Frisch's work.5 To Andy Schoene we owe thanks for calling our attention to ... A. T. Bharucha-Reid (New York: Academic Press, 1968).
RANDOM EVOLUTIONS, MARKOV CHAINS, AND SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS BY R. J. GRIEGO AND R. HERSH* UNIVERSITY OF NEW MEXICO, ALBUQUERQUE
Communicated by Einar Hille, December 5, 1968
Abstract.-Several authors have considered Markov processes defined by the motion of a particle on a fixed line with a random velocity 6, 8, 10 or a random diffusivity.5' 12 A "random evolution" is a natural but apparently new generalization of this notion. In this note we hope to show that this concept leads to simple and powerful applications of probabilistic tools to initial-value problems of both parabolic and hyperbolic type. We obtain existence theorems, representation theorems, and asymptotic formulas, both old and new. Notations.-For i = 1, ... n, Ti(t) is a strongly continuous semigroup of bounded linear operators on a fixed Banach space B. Ai is the infinitesimal generator of Ti. v(t) is a stationary MAlarkov chain with state space { 1, . n} and infinitesimal matrix Q = (qij). l is the n-fold Cartesian product of B with itself. f is a generic element of B; f = (fi), i = 1, .I. . n, a generic element of B. For a sample path w(t) of v, rj(w) is the time of the jth jump, and N(t,w) is the number of jumps up to time t. 'yj(t,w) is the occupation time in the jth state up to time t. Ei is the expectation integral on sample paths w, under the condition w(O) = i. Definition 1: A random evolution M(t,co) is the product
M(t)
=
Tv(o)(rl)TV(T,)(r2
-
TO) . .. TV(TN(t)) (t
-
TN ).
Definition 2: The "expectation semigroup" T is the (matrix) operator on B specified componentwise by (T(t)f)i = Ei[M(t)fv(g)], where the integral is taken in the sense of Bochner. Intuitively, a random evolution is one possible history of a machine capable of n modes of operation, subject to the control of a monkey who flips the control switch at random from one to another of n positions. The expectation semigroup is the average or expected outcome of this procedure. Before considering any concrete examples, we record three abstract theorems that are easy and basic. THEOREM 1. T(t) is a strongly continuous semigroup of linear operators. THEOREM 2. The Cauchy problem for an unknown n-vector il(t), t > 0, bt Ut
=
Ai1s +
qj
is solved by = T(t) 35(t)
305
i = f, i()
(1)
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Remark 1: With appropriate modifications in notation, we may allow v(t) and Ti(t) to be nonstationary, so that Ai and qi1 may depend on t in (1). Remark 2: Since B is an abstract space, our theorem applies both to the "pure initial-value problem" (B a function space on a domain without boundary) and to the "mixed initial-boundary value problem" (B a function space on a domain with boundary, subject to homogeneous, linear boundary conditions). THEOREM 3. If AiA, = AjAi for ij = 1,... n, n
Mt)f) i = Ei( I| Tj(,j(t))fv(0)). Applications: (i) Parabolic systems: In reference 4, it is shown that if a second-order linear Petrowsky-parabolic system has a fundamental solution matrix with only nonnegative elements, then, in the ith equation, only the ith unknown is differentiated; the coefficients of the other unknowns are nonnegative. (1) includes the most general system of this type, subject only to the inessential normalization 2;jqj = 0 and to the condition that Q is independent of x. We can take B to be the function space C(E), E a continuum in some Euclidean space En, and the T,(t) as the semigroups of n diffusion processes on E. We then have a stochastic solution of the most general type of parabolic system that can admit such a solution. (ii) One-dimensional hyperbolic systems: If Ai = ci(bl/x), (1) is a onedimensional first-order hyperbolic system. (Note that it appears already in diagonal (characteristic) variables as a consequence of its physical derivation.) This case has been studied" 10 for constant ci and - o < x < c. Since in this case Ti(t)f = f(x + cit), Theorems 2 and 3 give the solution formula ui(t) = Ei(fv(Z)(x + c3cj-y(t))) (iii) The Poisson case, and a generalized "telegraph" equation: If n = 2,
if A, = -A2 = A, and if Q = (order Cauchy problem
a), then u = ul + u2 solves the second-
Utt = A'u - 2a(t)ut, u(O) = fl + f2, ut(O) = A(f, - f2).
(2)
Now ul and u2 are given by Ei(T(y -Y2)fr(t)). For this Q, N(t) is a Poisson process with intensity a, and we get as a simple consequence of our general theory the following formula first obtained by Kac for the case A = b/bx. THEOREM 4. If w(t) is the unique solution of wtt = A w for t > 0, w(0) = f, Wt(0) = Ag, g and f in the domain of A then u(t) = El [w (f-1)N(8)ds)I solves utt = A u - 2aut, u(0) = f, ut(0) = Ag. In reference 8, this formula is derived heuristically for A = a/ax, and then verified a posteriori by computing moments. (See also ref. 9 for a simpler and more general verification.) The a posteriori verification then unexpectedly turns out to be valid for more
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general second-order equations wtt = Pw. Our derivation lays bare the probabilistic mechanism that brings about this surprise. It is simply a random alternation between T(t) and T-'(t) = T(-t), where the group T(t) is generated by the square root of P. In this connection, Seeley's work' on roots of elliptic operators can be used, together with Theorem 23.9.5 of reference 7, to bring under Theorem 4 quite general second-order hyperbolic equations. (iv) Asymptotic estimates, singular perturbations, and a Gaussian transformation: Finally, we apply Takacs'" central limit theorem for occupation times to get an easy new attack on a wide class of singular perturbation problems. If we replace a by a/E, A by A/v/c, then (2) becomes (3) Eutt = A2u - 2au . Now we find that Takacs' theorem enables us to evaluate the limit of u(t) as e\'U0. THEOREM 5. If u satisfi es for t > 0, eutte = A2uE - 2auge, uE(0) = f, ut'(O) = Ag, g and f in the domain of A 2, then, for all t > 0, uE(t) converges weakly to uP(t) = EN(O,tla) [T(s)f] as e \E 0. That is, uQ is obtained, as was uE, by applying to f the solution operators T(s) averaged over a random time 5; but in the limit this random time has a normal (Gaussian) distribution, with mean 0 and variance t/a. THEOREM 6. The solution of
ut= 2a A2u, (t > 0) u(O) = f
(4)
is given by u4 = EN(O,t/a) [T(s)f] = (2wrt/a)-'/1 f2_ T(s)f(x) exp (-as2/2t)ds. This formula is equivalent to Theorem 2 of reference 3, where it appears in a form that masks its probabilistic content. It is a transform formula that shows how to transform a given continuous group T(t) with generator A into a holomorphic semigroup with generator A2/2a. Theorems 5 and 6 together imply that solutions of (3) converge to solutions of
(4).
For A2 = (82/8x2), this result goes back to Hadamard. For A2 a self-adjoint second-order elliptic operator, the result was proved by Zlamal.14 A concrete theorem of this type for higher-order equations is proved in reference 10, where it is used to obtain a probabilistic limit theorem. Here, on the contrary, we use a central limit theorem to prove the singular perturbation result, under hypotheses no stronger than the solvability of (3) and existence of a closed, densely defined square root A. For example, if A = i2(b2/bx12), we conclude that the solution of the fourthorder parabolic equation ut + A2u = 0 is the expected value of random solutions (normally distributed in time) of the
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Schrodinger equation ut = iAu, and that it is the limit of solutions of (5) EUtt + U t + A u = 0 (vibrating beam equation with damping). The We of (5) is also the expected value of random solutions of Schr6dinger's equation, but with a probability distribution defined by the occupation times of a Poisson process with intensity (1/2e). (Since, in this example, A is a constant-coefficient partial differential operator, the fact that uW converges to u could also be obtained by Fourier methods; see Theorem 4 of Bobisud.2) In this work we benefited from the interest, advice, and encouragement of Professor L. H. Koopmans. To Peter D. Lax we owe the useful term, "random evolution." Joe Keller kindly told us of Frisch's work.5 To Andy Schoene we owe thanks for calling our attention to Theorem 23.9.5 in Hille-Phillips.? * This work was supported in part by NSF grants GP-8290 and GP-8856. 1 Birkhoff, G., and R. E. Lynch, "Numerical solution of the telegraph and related equations," in Numerical Solution of Partial Differential Equations (Proc. Symp. Univ. Md., 1965) (New York: Academic Press, 1966), pp. 289-315. 2 Bobisud, L., "Degeneration of the solutions of certain well-posed systems of partial differential equations depending on a small parameter," J. Math. Anal. Appl., 16, 419-454 (1966). 3Bragg, L. R., and J. W. Dettman, "Related problems in partial differential equations," Bull. Am. Math. Soc., 74 (1968). 4Chabrowski, J., "Les solutions non negatives d'un systbme parabolique d'equations," Ann. Polon. Math., 19, 193-197 (1967). 5 Frisch, U., in Wave Propagation in Random Media, Probabilistic Methods in Applied Mathematics, ed. A. T. Bharucha-Reid (New York: Academic Press, 1968). 6 Goldstein, S., "On diffusion by discontinuous movements and on the telegraph equation," Quart. J. Mech. Appl. Math., 4, 129-156 (1950). 7Hille, E., and R. Phillips, Functional Analysis and Semigroups (Providence, R. I.: Am. Math. Soc., 1957). 8 Kac, M., "Some stochastic problems in physics and mathematics," in Lectures in Pure and Applied Science (Magnolia Petroleum Co., 1956), no. 2. 9 Kaplan, S., "Differential equations in which the Poisson process plays a role," Bull. Am. Math. Soc., 70, 264-268 (1964). 10 Pinsky, M., "Differential equations with a small parameter and the central limit theorem for functions defined on a finite Markov chain," Z. Wahrs. Geb., 9, 101-111 (1968). 11 Seeley, R. T., "Complex powers of an elliptic operator," in Proceedings, Symposium on Pure Mathematics (Providence, R. I.: Am. Math. Soc., 1967), vol. 10, pp. 288-307. 12 Stratonovich, R. L., Conditional Markov Processes and Their Application to the Theory of Optimal Control (New York: Elsevier, 1968). 13 Takacs, L., "On certain sojourn time problems in the theory of stochastic processes," Acta Math. Acad. Sci. Hungar., 8, 169-191 (1957). 14 Zlamal, M., "On a singular perturbation problem concerning hyperbolic equations" (University of Maryland: Institute for Fluid Dynamics and Applied Mathematics Lecture Series, 1965), no. 45.
