Betz, C.

Gzyl, H.

English

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stichting mathematisch centrum AFDELING ZUIVERE WISKUNDE (DEPARTMENT OF PURE MATHEMATICS) C. BETZ & H. GZYL zw 187/83 PROBABILISTIC APPROACH FOR COMPARING FIRST EIGENVALUES ~ MC APRIL kruislaan 413 1098 SJ amsterdam tlltil!OfHEEK M/\TH:i:~/i:\TtSCH CENTHUM -AMS:Tri,,hM-Printed at the Mathematical Centre, Kruislaan 413, Amsterdam, The Netherlands. The Mathematical Centre, founded 11 February 1946, is a non-profit institution for the promotion of pure and applied mathematics and computer science. It is sponsored by the Netherlands Government through the Netherlands Organization for the Advancement of Pure Research (Z.W.O.). 1980 Mathematics subject classification: 60J 99-35B99 Copyright© 1983, Mathematisch Centruni, Amsterdam Probabilistic approach for comparing first eigenvalues by . • B *) H k G 1 *) Cr1.st1.na etz & enry zy ABSTRACT We use probabilistic subordination and time change to compare the first eigemralues for problems of the type Gijl + AljJ = 0; GljJ - qljJ + :\pljJ = 0, with zero boundary conditions. KEY WORDS & PHRASES: First eigenvalue, subordinate process, time change. Permammt adress: Apartado 52120. Caracas 1050-A Venezuela. Work done during a sabbatical stay at the Mathematical Centre. I • INTRODUCTION The aim of this paper is to use probabilistic subordination and time change to compare the first eigenvalues of the problems (1. 1) G¢ +A¢= 0 in D ¢/oD = 0 (1. 2) G¢ - q¢ + ;\¢ = 0 in D ¢/aD = 0 ( 1 • 3) G¢ - q¢ +AP¢= 0 in D ¢/aD = 0 for an adequate class of operators G. In Section 2 we consider a Hunt process (Xt) with state space (E,E) whose semigroup possesses a nonnegative symmetric density f(t,x,y) with respect to a Radon measure m on (E,E), such that if D is a bounded open subset of E with m(oD) = 0 then ff f 2 (t,x,y)m(dx)m(dy) < ~ DD and such that, if T = inf{t > 0 then Vx ED. Xt E Dc} is the first exit time from D, By stopping (Xt) at the boundary of D and subjecting it to a local death rate q(x) we are able to compare the first eigenvalues of the problems (1.1) and (1.2), when G is the infinitesimal generator of (Xt). In Section 3 we consider an adequate time change of the process with generator G-q to compare the first eigenvalues of problems (1.2) and (1 .3). Basic for all the comparison results is the fact that under the stated assumptions on the semigroup one can get eigenvalue expansions for these various processes which allow one to give a simple characterization of the first eigenvalue. This characterization (proposition 1 in Section 2) is well known for diffussions (see chapter 14 of [4]). Section 4 is devoted to the discussion of examples. We refer the reader to [l] for the basic notation and definitions 2 throughout the paper. 2. COMPARISON OF (1 .1) AND ( 1. 2) Let (X~) be the process (Xt) killed when it leaves D and (Q~) its transition semigroup, that is It is proved in [SJ that there is a sequence {A.} of eigenvalues of G such J that Os Al s A2 ~ ••• s Ans •.• , An+ m and a complete orthonormal set of ~m -A ·t eigenfunctions {~j} such that the series lj=l e J ~j(x)~j(y) converges absolutely and (2.1) -At f Q~f(x) =le n ~n(x) n ~ (y) f (y)m(dy) n D for f e: L 2 (D). A similar expansion can be found for the semigroup ~ x D Qtf(x) = E [f(Xt)Mt;t<TJ of the process subordinate to (Xt) with respec~ to the multiplicative functional Mt= exp(-J~ q(Xs)ds) where q is a con-tinuous positive function on E (see also [SJ). The first eigenvalue of the problem (1. 1) G$ +AW= 0 in D, $/3D = 0 can now be characterized as follows PROPOSITION 1. If Al is the first eigenvalue of the problem (1.1) then X AT Al= sup{A: sup E [e J < m}. xe:D PROOF. It follows from (2.1) that ~ (y)m(dy). n 3 Let Kn= JD $n(y)m(dy) then one has 00 00 Ex[eAT J - J ;u x l J --(A -A) t n = e dP (T>t) = K $ (x) A e dt = n n n 0 n 0 = I K $n(x)(AA~A). n n n Clearly, when A+ Al this series diverges. Also, EJyeATJ is an increasing function of A, so it suffices to show that Ex[eATJ < oo for A< A1, Vx ED. Now, if O <A< Al then An+l A ---< _n_ An+l-A - An-A' hence REMARK 1. A similar characterization for the first eigenvalue of problem (1.2) clearly holds REMARK 2. The heuristics of the situation is as follows: 00 00 Ex[eAT] = - J eAtdPx(T>t) = J AeAsPx(T>s)ds = 0 0 00 00 = J AeAS Ex[l(Xs),s<T] = A J 0 eAS QDsl (x) h x( ) 1 wen P T<oo = 0 So what one does is "look at the poles of the resolvent" In order to compare the first eigenvalues of problems (1.1) and (1.2) we consider (Xt), the canonical realization of the process subordinated f t ~ to (Xt) with respect to Mt= exp(- 0q(Xs)ds) on the space n = nx[0, 00 ], and ~ ~ denote by P its corresponding measures on n (see [1] ch.3 for all the con-s true tions) • ~ Following [1], let y: n + [0, 00 ] be the projection y(w,A) = A, and = ]. t > ~ T 'nf{ 0 Xt De} 4 ~ C T = inf{t > 0 : Xt ED}. ~ LEMMA 1. ~[eATJ :s; Ex[eATJ, Vx ED. PROOF. It follows from the definition of (Xt) that {T>t} = {x s ED, s :s; t} = {X En, s :s; s t, y > t} = = {T > t; y > t} and therefore, by the definition of p'X we have ~ ~ X P (T>t) = E [Mt;T>t]. Now (X) (X) Ex[eATJ J e At dp'X (T>t) J At X = = e dE [Mt;T>t] = 0 0 T T = I + AEx J eA~tdt :s; 1 + AEx J eAtdt = Ex[eATJ. 0 0 the integration by parts in the third equality is possible since Px(T<<x>) = I Vx. ED. The fol11owing comparison result now follows easily from Proposition I and Lemma 1. THEOREM 1 • Le·t A 1 be the smaUest eigenvalue of problem (1 .1) and 5: 1 the smaUest eigenvalue of problem (1 .2) then REMARK 3. In the same manner, one can compare the first eigenvalues of the problems G~ - q1~ +A~= o in D ~/an= o G~ - 4 2~ + µ~ = o in n ~/an= o for q 2 ~ q1 • In fact, let G-q 1, and l1et (X ) be the (Xt) now denote the process with generator process subordinated to (Xt) with respect to t t Mt= exp(- J0 (q2(xs)-q 1(xs))ds) then, the same reasoning as before gives 5 REMARK 4. The comparison results will hold whenever a characterization for the principal eigenvalue, as the one given in proposition 1 holds; there-fore, these results will be valid for the elliptic operators considered by FRIEDMAN [4] Vol.2, Chapter 14. 3. COMPARISON OF (1.2) AND (1.3) We will now compare the first eigenvalues of the problems ( 1 • 2) (G-q)$ +A$= o in D $/an= o (1 .3) (G-q)$ +AP$= 0 in D $/an= 0 for p continuous and p ~ c > O. ~ Let (Xt) now be the process with generator G-q = G (we warn the reader that to be consistent with the notation of section 2 we should work with (Xt), but w1e feel this would become too confusing). Suppos1e (X ) has a symmetric transition density p(t,x,y) with respect t to the Radon measure m which satisfies (3. 1) (3. 2) J f 2 p (t,x,y)m(dx)m(dy) < ~ D D J p(t,x,x)p(x)m(dx) < ~ Vt ]) for bounded sets D. Let (At) be the following additive functional of (Xt) t At= J 0 p (X )ds s 6 and let (Lt) be the right continuous inverse of (At); that is Lt= inf{u; Au> t}. One may check that (Lt) satisfies t Lt = f p (;s ) , L s (X ) being the time changed process. Lt Call Xt = XLt' then .., ... X X = (n,F,FLt'Xt,eLt'P) is a strong ~rkov process (se~ [1] Ch.V) and it follows from Dynkin's formula that (Xt) has generator;-The following proposition allows one to use the same procedure as that of proposition 1 in Section 2 to characterize the first eigenvalue of problem (1 .3) as ... X AT Al= sup{A: sup E [e J < m} xe:D where ... .., C T = inf{t > 0: xt e: D }. PROPOSITION 1. Under the assumptions (3.1) and (3.2) and the symnetl'y of p(t,x,y); the proaess (Xt) possesses a symmetl'ia transition density p(t,x,y) with respeat to the measure p(y)m(dy) and moreover, if Dis bound-ed then (3.3) f j p2(t,x,y)p(x)p(y)m(dx)m(dy) < = DD PROOF. Observe that u0 f ex> where 7 If (Xt) has a symmetric transition density p(t,x,y) then (Q~) has a sym-metric transition density as well (see [5]), call it qa(t,x,y). It is easy to see that (3.4) a -act q (t,x,y) ~ e p(t,x,y) (recall c is such that p(x) ~ c > 0). O f 00 a Let va(x,y) = 0 q (s,x,y)d~ then by inverting Laplace transforms we get, for an appropriate contour y i\f(x) = 2!i j e0 t u0 f (x)dcr = y = j (2!i j e0 t v~(x,y)dcr)f-(y)p(y)m(dy) y which proves the existence of the symmetric transition density p(t,x,y). ... 1 j at O p(t,x,y) = -2 . e v (x,y)dcr. 1r1 a y To prove (3.3) it is enough to show that j p(2t,x,x)p(x)m(dx) < 00 • D But, for all a> 0 00 j e-atp(t,x,y)dt = v~(x,y) 0 00 00 ~ f 0 -act e p(t,x,y)dt = J -at t = C e p(c,x,y)dt. 0 8 Hence p(t,x,y) ~ cp(.!.,x,y) C and the result follows from (3.2). Let then (3.5) and so (3 .6) T = inf{t > 0: xt e DC} i = inf{t > 0: it E DC} .... T = A(T) .... {T > t} = {A(T) > t} = {T > Tt} (see [ 1] ch.V). The foll,owing analytic lemmas will be needed for the comparison result. LEMMA 1. Let :F(t,w) be a measurabl,e function on lR. x n, ((n,F,P) being a proobabil.ity space) such that t -+ F(t,w) is monotone and roight continuous faro each w. Then, faro f ~ 0 f f(t)dE(F(t,w)) = E f f(t)dF(t,w). (E denotes expectation with respect to P). PROOF. Easy. LEMMA 2. Let£: lR -+ lR be 1-1 and differoentiable. If o(x-a) denotes the delta function at a then f -1 g(x)o(f(x)-a)dx = g(f _Ca)) . f' (f 1 (a)) PROOF. Easy by choosing an approximating o-sequence (h). n 9 THEOREM 1. If p < 1 tJzen the first eigenvalue of problem (l .3) is bigger than the first eigenvalue of problem (l .2) and if p > 1 the first eigen-value of problem (1.3) is smaller than the first eigenvalue of problem (1.2). PROOF. It suffices to compare Now 00 00 .... = - J - f Ex[/'TJ /1.t dPX(T>t) At X = e dP (T>T ) • t 0 0 Using lemmas 1 and 2 we get 00 At x AT e p(XA )o(t-T)dt = E [e p(XA )] T T the theorem now follows easily. 4. EXAMPLES In this section we list some situations in which the comparison re-sults hold. In particular, we see that the classical results of examples 1 and 2 (see [3] ch.VI) are valid for more general situations. EXAMPLE 1. The eigenvalues of the problem 6$ - q$ +A$= 0 inn,$= 0 on an; increase when q increases. EXAMPLE 2. The eigenvalues of 6$ - q$ +A~= 0 inn,$= 0 on an; increase when the domain n decreases. EXAMPLE 3. This is a generalization of example 1. Let G be the Laplace-Beltrami operator on a Riemannian manifold or compact Lie group and nan open bounded subset. The same conclusions hold. See [7] or [8] for the 10 appropriate constructions. EXAMPLE 4. The following construction, going back to PHILLIPS [6] yields a larger class of processes (and eigenvalue problems) for which the com-parison results apply. Suppose (Xt) has a transition density p(t,x,y) such that p(t,x,y) = r e n -At n where the(~) are an orthonormal, complete set, and(~ ,A) are all the n n n solutions to G~ +A~ = 0 plus appropriate boundary conditions. n n n Let (t,t) t~O now denote a subordinator (a process with stationary in-dependent increments on [0, 00 ]) independent of (Xt). Define a new semigroup on the same state space by 00 00 f I -Au f Qtf(x) = Puf(x)Tt(du) = l ( e n Tt(du))~n(x) ~n(y)f(y)dy 0 n O -µ t f =le n ~ .(x) ~ (y)f(y)dy n n n (3.1) where µn = aAn - f (e-Anu_l)v(du), (a,v) being the characteristics of {Tt). In this case, the process (XTt'n,Px) where Px is constructed from T T (Qt) via the Kolmogorov extension theorem, has transition semigroup (Qt) and infinitesimal generator (3.2) G f(x) = aGf(x) T. 00 - J (P f(x)-f(x))v(du). u 0 One has G ~ + µ ~ = 0 and, from completeness, there are no more solutions T n n n to G w +AW= 0 (with the same boundary conditions as for Gw +AW= 0). T EXAMPLE 5. The comparison results can be applied to the synnnetric stable process in Rn of index a, 0 < a :;;; 2; that is, the process with stationary independent increments whose continuous transition density relative to Lb . lRN. e esgue measure in is t > 0. 11 See [2J and [5] in connection with this example. It is interesting to observe the connection with example 4, which is as follows: If (N:) is the one sided stable semigroup in (0,m), BE (0,1) and Pt(x,A) the transition semigroup of the Brownian motion process in lRN then, the semigroup of the synnnetric stable process in lRN satisfies m (see [1] ch.I;2.20). If one now considers the Brownian motion process killed when it first reaches Dc and one performs on it the time change of example (4) with (,t) being (N~/2) one obtains, as might be expected, the synunetric stable process of index a, killed when it first leaves D. This fact follows from the fol-lowing two lemmas, which might also be interesting for other situations. Following the notation of example 4 define T T = inf{t > 0: X 't € DC} T inf{t > 0: C = Xt ED} lt = inf{s > 0: 's ~ t} the left continuous inverse of (Tt). It is well known that {l > t}.,.. {. < s} s t and therefore {l(T) > t},... {,t < T}. LEMMA 1 • l {T) = T • T PROOF. T, ~ t..,.. X.s ED, Vs< t.,. 's < T, Vs< t..,. s < l(T) Vs< t ..,. t s l{T). If E[ J denotes expectation with respect to the distribution of 't then one has as an immediate consequence of Lemma 1 and the remarks preceding it, that 12 LEMMA 2·. E[Ex[f(X ); •t < T]] = E[Ex[f(X ); t < T ]]. •t •t T EXAMPLE 6. The Ornstein-Uhlenbeck process in lR ~as a transition density f(t,x,y) with respect to the measure m(dy) = e-y /2 dy which is symmetric see [2]. EXAMPLE 7. The following situation may apply to discretized versions of examples 1 or 2. Consider a synunetric Markov chain on a lattice, with . transition among nearest neighbors only. Let D be a subset of the state space and let an denote the subset of Dc consisting of nearest neighbors to points in D. If Q denotes the rate matrix of the process killed on leaving D, then our comparison r,esult applies to the problems REFERENCES l Q(i,j)$(j) + A$(i) = o; $(i) = o, i E an. jED I Q(i,j)$(j) - q(i)$(i) + A$(i) = O; $(i) = O, jED i E an. [1] BLUMENTHAL, R.M. and R.K. GETOOR, Markov proaesses and Potential theory, Academic Press, New York (1968). [2] BLUMENTHAL, R.M. and R.K. GETOOR, The asyrrrptotia distribution of the eigenvalues for a aZass of Markov operators, Pacific Jour. Math. Vol. 9 (1959) pp.399-408. [3] COURANT, R. and D. HILBERT, Methods of MatherrrztiaaZ physias, Vol. I. Interscience - New York, (1953)-[4] FRIEDMAN, A., Stoahastia differential equations and appZiaations. Academic Press, New York (1976). [5] GETOOR, R.K., Markov operators and their assoaiated semi-groups, Pacific Jour. Math. (1959) Vol. 9 pp.449-472. [6] PHILLIPS, R.S., On the generation of semig:roups of "linear operators, Pacific Jour. Math. 2 (1952) pp.343-369. [7] PINSKY, M.A., Isotropic transport process on a Riemannian manifoZd, T.A.M.S. Vol. 218 (1976) pp.353-360. [8] STEIN E.M., Topics in ha.rmonic anaZysis, Annals of Math. Studies #63. Princeton Univ. Press. Princeton N.J. (1970). 13 1 1 EI 

Probabilistic approach for comparing first eigenvalues

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