One-Dimensional Diffusion Operators

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1 One-Dimensional Diffusion Operators Stanley Sawyer Washington University Vs. July 7, Basic Assumptions. Let sx be a continuous, strictly-increasing function sx on I, 1 and let mdx be a Borel measure mdx on I that is, mk < for compact subsets of K I. Then we can define a Feller-type operator Lfx d d fx 1.1 dmx dsx for appropriate functions fx on I. Here sx is called the scale function of Lfx and mdx is the speed measure. See e.g. Feller 1952, 1954, 1955, Ito and McKean 1965, Sawyer Assume further that sx C 2 I CI, s x >, sdx s xdx mdx mxdx, mx >, mx CI 1.2 for < x < 1 and I [, 1]. Since sx CI, s and s1 are finite. By CS we mean the set of all continuous functions on S and by C k S for an open set S the set of all k-times-continuously differentiable functions on S. Then Lfx 1 mx d dx 1 s x f x if fx C 2 I. In the following we also assume that 1.3 /2 sy s mdy <, s1 sy mdy < 1.4 1/2 Condition 1.4 is called the condition that the operator Lfx has exit boundaries at and 1. The motivating example that we have in mind is and more generally sx x and mdx 2dx x1 x sx 1 e 2γx 2γ and mdx 2e2γx dx x1 x 1.5

2 One-Dimensional Diffusion Operators for γ. We leave it as an exercise to show that 1.3 and 1.5 correspond to the differential operator Lfx 1/2x1 xf x + γx1 xf x 2. An Integral Equation for Lfx. Define sx y s s1 sx y gx, y s1 s s1 sx sy s s1 s s1 sy sx s s1 s y x 1 x y where x y min{x, y}, and x y max{x, y}. Then since sx is increasing gx, y gy, x 2.2 s1 s gx, x min{s1 sx, sx s} 4 gx, y min{gx, x, gy, y} Define the linear operator Kfx gx, yfymdy s1 sx s1 s + x sx s s1 s sy s fymdy x s1 sy fymdy 2.3 for fx L 2 dm { fx : fx2 mdx < }. By Cauchy s inequality gy, yhy mdy gy, y 2 mdy hy 2 mdy where gx, y gy, y C and gy, ymdy < by 1.4 and 2.2. Thus by dominated convergence each Kfx C I where C I { hx CI : lim x hx lim x 1 hx } 2.4

3 One-Dimensional Diffusion Operators The following theorem shows that the integral operator Kfx in 2.3 is the inverse of the differential operator Lfx in 1.1 and 1.3 with zero boundary conditions that is, Lf C I. Theorem 2.1. If fx L 2 dm, then hx Kfx is the unique function hx C 1 I C I such that h x is absolutely continuous in I and Lhx fx for mdx-almost every x I. If also f CI, then h C 2 I and L x hx fx for all x I. Proof. Assume hx Kfx in 2.3. Thus hx s1 sx s1 s + x sx s s1 s sy s fymdy x s1 sy fymdy As above, h CI with h h1. Also, hx is absolutely continuous in x with h x s s1 sy x x s1 s fymdy x sy s s1 s fymdy 2.5 for almost every x. Since the right-hand side above is continuous, h x extends to a continuous function on CI such that both h x and h x/s x are themselves absolutely continuous. Moreover, d/dx h x/s x fxmx for almost every x. Thus by 1.3 Lhx fx for mdx- almost every x I. If f CI, then h C 2 I and L x h f by the same proof. Conversely, suppose that h 1 x C 1 I C I, h 1x is absolutely continuous, and Lh 1 x fx for almost every x. I now claim that h 1 h Kfx. First, let gx hx h 1 x. Then, following the same convention as in 2.5, gx C 1 I C I, g x is absolutely continuous, and L x gx Lhx Lh 1 x. In generation, the relations 1.3 and g C 1 I with g x absolutely continuous implies x g x g x + s x L x gymydy x 2.6 for < x, x < 1. Thus g x if L x gx for mdx-a.e. x and gx is constant. Since g g1, we must have g h h 1 and h h 1, which completes the proof of Theorem 2.1.

4 One-Dimensional Diffusion Operators We have actually proved a little more: Lemma 2.1. Assume gx C 1 I CI, g g1, g x is absolutely continuous, and Lgx 1 d g x mx dx s λgx 2.7 x for mdx-a.e. x and some constant λ. Then either gx for < x < 1 or else λ >. Proof. By 1.3, 2.6 holds with L x gy λgy for arbitrary < x, x < 1. If λ, then g x for a.e. x and gx g g1. Now assume λ. Assume max <x<1 gx gx > and λ <. Then 2.6 implies that g x > for x < x < x + ɛ for some ɛ >, which is inconsistent with the assumption that x is an interior maximum of gx. Thus λ >. Lemma 2.2. Assume fx L 2 I, dm and Kfx gx, yfy mdy λfx a.e. dm 2.8 for some constant λ. Then either fx a.e. dm. or else λ >. Proof. By Theorem 2.1 and Lemma Kfx is a Hilbert-Schmidt Operator. That is, I claim Theorem 3.1. For gx, y in 2.1 Proof. By 2.2 and 1.4 Thus kx gx, y 2 mdxmdy < 3.1 gx, y 2 mdy gx, x gx, y 2 mdxmdy gy, y mdy < gy, y mdy < kxmdx

5 One-Dimensional Diffusion Operators We can use a similar argument to obtain Theorem 3.2. Suppose f n, f L 2 I, dm and lim n and Kfx is given by 2.3. Then fn y fy 2 mdy lim Kf nx Kx n uniformly in x Proof. Then Kf n x Kfx 2 kx gx, y 2 mdy gx, y f n y fy mdy fn y fy 2 mdy fn y fy 2 mdy 2 uniformly in x since kx is bounded by Eigenfunctions and Eigenvalues of Kfx. Since Kfx is a symmetric Hilbert-Schmidt operator, or equivalently since gx, y is a Hilbert- Schmidt kernel in L 2 I, dm by Theorem 3.1, there exist eigenfunctions φ n x of K in L 2 I, dm Kφ n x gx, yφ n y mdy µ n φ n x 4.1 for nonzero real eigenvalues µ n satisfying the following conditions see e.g. Riesz and Nagy, 1955, p242. The functions φ n x are orthonormal, that is where φ n, φ m f, g φ n xφ m x mdx δ mn 4.2 fxgx mdx 4.3 for f, g L 2 I, dm, and any f L 2 I, dm can be written f g + c n φ n 4.4

6 One-Dimensional Diffusion Operators where Kg, g, φ n for all n, and the series converges in L 2 I, dm Riesz and Nagy ibid.. By Lemma 2.2 with λ, Kg implies g, so that g in 4.4. Thus the functions φ n x are complete orthonormal in L 2 I, dm and c n f, φ n in 4.4. That is, the expansion 4.4 with g holds for any f L 2 I, dm with c n f, φ n. Similarly, the eigenvalues µ n > by Lemma 2.2. By Theorem 2.1, the relations implies φ n x C 2 I C I, φ n φ n 1, and φ n x µ n Lφ n x. Thus 4.1 is equivalent to L x φ n x λ n φ n x, λ n 1/µ n, φ n φ n We can use the Hilbert-Schmidt property 3.1 and the properties of φ n x to find out more information about the eigenvalues µ n. Theorem 4.1. The function gx, y in 2.1 satisfies gx, y µ n φ n xφ n y 4.6 with convergence in L 2 I 2, dm 2. Moreover gx, y 2 mdxmdy µ 2 n < 4.7 Proof of Theorem 4.1. Since the functions φ n x in 4.1 or 4.5 are complete orthonormal in L 2 I, dm, the functions {φ m xφ n y} are complete orthonormal in L 2 I 2, dm 2 where I 2 I I and dm 2 dxdy mdxmdy. This means that if kx, y is any measureable function on the unit square with kx, y2 mdxmdy <, then kx, y c mn φ m xφ n y 4.8 m1 with convergence in L 2 I 2, dm 2 and Moreover c mn c 2 mn m1 gw, zφ m zφ n wmdzmdw 4.9 kx, y 2 mdxmdy < 4.1

7 One-Dimensional Diffusion Operators Since gx, y kx, y is in L 2 I 2, dm 2 by 3.1, gx, y satisfies 4.8 in the same sense. Since gx, yφ n ymdy µ n φ n x and φ n, φ m δ mn, the Fourier coefficients in 4.9 are c mn µ n δ mn. This implies 4.6 and 4.7. Theorem 4.2. Under the assumptions φ n x λ n C 1 gx, x 4.11 for an absolute constant C 1. If in addition gx, x mdx <, φ n x C 2 λ 2 n gx, x 4.12 Proof. By 4.1 and Cauchy s inequality φ n x λ n gx, y 2 mdy φ n y 2 mdy λ n gx, x gy, ymdy 4.13 since gx, y min{gx, x, gy, y} by 2.2. The right-hand side of 4.13 is finite by 1.4. Similarly by 4.1 and 4.11 by 2.2. We also have Theorem 4.3. φ n x λ n gx, y φ n y mdy The series C 1 λ 2 ngx, x gy, ymdy gx, y µ n φ n xφ n y 4.14 converges uniformly absolutely for x, y 1. In particular by 4.14 gx, xmdx µ n φ n x 2 mdx µ n < 4.15

8 One-Dimensional Diffusion Operators Proof. This is a special case of Mercer s Theorem Riesz and Nagy, 1955, p245. We give the proof for completeness. First, consider the remainder g n x, y gx, y n µ k φ k xφ k y k1 kn+1 µ k φ k xφ k y 4.16 with convergence in L 2 I 2, dm 2. Since the φ n x C I, the function g n x, y is continuous in I 2 and satisfies fxfyg n x, ymdxmdy kn+1 µ k f, φ k by I claim that this implies g n x, x for all n and x. For, if g n x, x < for some x I, then g n x, y < for x, y x ɛ, x + ɛ for some ɛ >. Let fx L 2 I, dm be any positive continuous function with support in the interval x ɛ, x + ɛ. Then fxfyg n x, ymdxmdy x +ɛ x +ɛ x ɛ x ɛ fxfyg n x, ymdxmdy < which contradicts It follows that g n x, x and n µ k φ k x 2 gx, x 4.18 k1 for x 1. Next, by Cauchy s inequality n µ k φ k xφ k y n µ k φ k x 2 n µ k φ k y 2 km km km n µ k φ k x 2 gy, y 4.19 km by This implies that, for each fixed x, the series Bx, y µ n φ n xφ n y

9 One-Dimensional Diffusion Operators converges uniformly in y. Recall that Bx, y gx, y a.e. dm 2 in the square by Theorem 4.1, but we will need pointwise identity. For any fy C c I that is, a continuous function of compact support K I, the series Bfx Bx, yfymdy µ n φ n xf, φ n converges numerically for each x. By Theorem 3.2 Kfx gx, yfymdy µ n φ n xf, φ n with uniform convergence in x. It follows that Bx, y gx, y fymdy for every x and f C c I. Since we can take fy Bx, y gx, yψy where ψy and ψ C c I, we conclude Bx, y gx, y for all x, y. Thus gx, x Bx, x µ n φ n x for all x with < x < 1, with 4.2 being trivial for x and x 1. Since both gx, x and φ n x are continuous on the closed interval [, 1], and since the convergence in 4.2 is monotonic since µ n > and φ n x 2, it follows from Dini s theorem that the convergence in 4.2 is uniform in x. It then follows from 4.19 that the series 4.14 converges uniformly in I 2, which completes the proof of the theorem. A quick proof of Dini s Theorem: For each x [, 1], the series converges within 2ɛ > for n n x, and hence within ɛ > for n n x and y x < δ x and y I for some δ x >. Note [, 1] x x δ x, x + δ x and use the Heine-Borel theorem. 5. Diffusion Semi-Groups and Non-negativity. Define pt, x, y e λnt φ n xφ n y 5.1 for < t < for the eigenvectors and eigenvalues φ n x and λ n defined in 4.5 and 4.1. Since φ n x C 1 λ n by 4.11 where 1/λ n <, the

10 One-Dimensional Diffusion Operators series converges uniformly for x, y 1 and a t < for any a >. By Mercer s Theorem Theorem 4.3, 1/λ n φ n xφ n y converges absolutely uniformly for x, y 1. Thus pt, x, ydt gx, y 5.2 with uniform convergence of the integral for x, y 1. Since the series 5.1 converges in L 2 I, dm for each x, pt + s, x, z pt, x, yps, y, zmdy 5.3 for t, s >. Similarly, define Q t fx It follows from 5.3 that The main result of this section is pt, x, yfymdy, f L 2 I, dm 5.4 Q t+s fx Q t Q s fx, s, t > 5.5 Theorem 5.1. For pt, x, y in 5.1 and t >, x 1, i pt, x, y y 1 ii pt, x, ymdy 1 iii For any f C I, lim t pt, x, yfymdy fx 5.6 uniformly for x 1. Proof. Choose hx C 2 I such that, for some a >, hx for x a and 1 a x 1 and set fx L x hx. Then f CI also satisfies fx for x a and 1 a x 1. Since µ n 1/λ n in Theorem 4.3 Mercer s theorem, the series rt, x, y e λ nt λ n φ n xφ n y 5.7

11 One-Dimensional Diffusion Operators converges uniformly for x, y 1 and t <. Also, r, x, y 1/λ nφ n xφ n y gx, y. Since mdy mydy where my is bounded and continuous for a y 1 a, the function ux, t rt, x, yfymdy 5.8 e λ nt φ n xφ n y fymdy e λ nt λ n λ n φ n x φ n yfymdy with uniform convergence for x 1 and t <. Also 1 1 ux, φ n x φ n yfymdy λ n r, x, yfymdy gx, yfymdy hx by 4.14 and Theorem 2.1. Thus by 5.7, 4.15, and 4.5, the function ux, t in 5.8 is continuous for x 1 and t and satisfies t ux, t L xux, t, < x < 1, < t < 5.9 u, t u1, t, t < 5.1 ux, hx, x 1 It follows from 4.1 and gx, y gy, x that Hence by 5.8 φ n xhxmdx ux, t 1 λ n φ n xgx, ymdx fymdy φ n yfymdy e λnt φ n x φ n yhymdy e λ nt φ n xφ n y hymdy pt, x, yhymdy 5.11

12 One-Dimensional Diffusion Operators By maximum principles for parabolic linear partial differential equations, any solution of 5.9 that is continuous for x 1 and t T satisfies max ux, t 5.12 { } max max u, t, max u1, t, max ux, t T t T x 1 x 1, t T with a similar inequality for the minimum. See Lemma 3, p167, of Protter and Weinberger, If hx 1 in 5.1, this implies ux, t 1 for x 1 and < t <. We have now shown that ux, t in 5.11 satisfies 5.12 for any h C 2 I with hx for x a and 1 a x 1 some a >. If either pt, x, y < or pt, x, ymdy > 1 for any t > and < x, y < 1, we quickly arrive at contradiction. This completes the proofs of parts i and ii of Theorem 5.1. If fx is smooth and vanishes near the endpoints, part iii of Theorem 5.1 follows from 5.11 and the arguments from 5.8 to 5.1. By part ii Q t fx pt, x, y fy mdy f max fy 5.13 y 1 for f C[, 1]. The uniformity in 5.13 implies that if 5.6 holds for all h is a subset L C I, then it also holds for all h L with closure in the uniform norm on, 1. Thus 5.6 follows for all f C I. This completes the proof of Theorem Hille-Yoshida Theory. The linear operators Q t defined in 5.4 satisfy Q t+s Q t Q s by 5.5 and lim Q tf f for all f C I 6.1 t by 5.6 where f max x 1 fx. In other works, Q t is a stronglycontinuous semi-group of linear operators on the Banach space B C I. The infinitesimal generator of a strongly continuous semigroup of linear operators on a Banach space B is defined to be the linear operator on the subspace D DA B defined by DA { h B : lim t 1/tQ t h h f for some f B } 6.2 with Ah f. The operator A is linear on DA, but generally DA B. In general DA B if Q t is strongly continuous, since, for any a > and f B, h a Q sfds DA with Ah Q a f f.

13 One-Dimensional Diffusion Operators If Q t f C f for all f B and any t > and C <, then exists constants C 1 > and real K such that Q t f C 1 e Kt f 6.3 for all t < and f B. If λ > K, the resolvent operator R λ f is then a bounded linear operator on B. e λt Q t fdt 6.4 Lemma 6.1. Let Q t be a semigroup of linear operators on a Banach space B satisfying 6.1 and 6.3. Then, for any λ > K for K in 6.3, R λ maps B onto DA and λh Ah f whenever h R λ f Proof. If h R λ f for f B, then Q t h e λs Q t+s fds e λt e λs Q s fds t e λt h e λt e λs Q s fds Thus 1/hQ t h h λh f as t. This implies that, for any f B, h R λ f DA and Ah λh f, or λ Ah f. In particular, i the range of R λ is contained in the domain DA, ii the range of λ I A is all of B, and iii λ AR λ f f for all f B. Conversely, assume h 1 DA. Set f λ Ah 1 and h R λ f. Then λ Ah f λ Ah 1 and, if g h h 1, g DA and λ Ag. In general if g DA and s >, then 1/tQ t IQ s g 1/tQ t+s Q s g Q s 1/tQt g g Q s Ag as t. This implies that Q s g DA and AQ s Q s A. If Ag λg, then Q t g DA for all t > and AQ t g Q t Ag λq t g. This implies Q t g e λt g, which violates 6.3 if g and λ > K. Thus g, which implies h 1 h. Hence the range of R λ is all of DA, which completes the proof of the lemma. We can find A and DA exactly for the semigroup Q t defined by 5.4 in Section 5 for B C I. t

14 One-Dimensional Diffusion Operators Theorem 6.1. Let L x d/dmx d/dsx be the operator in 1.1 and let Q t be as in 5.4 for B C I. Then with Proof. DA { h : h C 2 I C I and L x h C I } 6.6 Ahx L x hx d d dmx dsx hx It follows from 5.1, 4.11, and Theorem 5.1 that Q t fx pt, x, yfymdy C λ 2 ne λnt f for f CI and t <. Since ψt λ2 ne λ nt /e λ1t is bounded and continuous for 1 t < with lim t ψt C where C is positive and finite, it follows that Q t fx Ce λ 1t f for t <. Thus 6.3 holds with K λ 1. Hence by Lemma 6.1 with λ R fx Q s fsds gx, yfymdy by 5.2. Hence DA is the range of R on B C I, which by Theorem 2.1 is exactly DA in 6.6. By Lemma 6.1, h DA if and only if h R f for some f B with Ah f, so that Ah L x h by Theorem 2.1. References. 1. Feller, W The parabolic differential equations and the associated semi-groups of transformations. Annals of Mathematics 55, 2nd Ser., Feller, W The general diffusion operator and positivity preserving semi-groups in one dimension. Annals of Mathematics 6, 2nd Ser., Feller, W On second order differential operators. Annals of Mathematics 61, 2nd Ser., Ito, K., and J. P. McKean, Jr 1965 Diffusion processes and their sample paths. Springer-Verlag.

15 One-Dimensional Diffusion Operators Protter, Murray, and Hans Weinberger 1967 Maximum principles in differential equations. Prentice Hall, New Jersey. 6. Riesz, Frigyes, and Bela Sz.Nagy 1955 Functional Analysis. Frederick Ungar Publishing, New York 6th printing, Sawyer, S. A A Fatou theorem for the general one-dimensional parabolic equation. Indiana Univ. Math. J. 24,