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A Summary of Dirichlet Form Theory

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A Summary of Dirichlet Form Theory A Summary of Dirichlet Form Theory Our treatment in this appendix follows that of the standard reference [72] – see also, [106, 3]. A.1 Non-Negative Definite Symmetric Bilinear Forms Let H be a real Hilbert space with inner product p¨, ¨q. We say E is a non- negative definite symmetric bilinear form on H with domain DpEq if •DpEq is a dense linear subspace of H, •E : DpEq ˆ DpEq Ñ R, •Epu, vq “ Epv, uq for u, v P DpEq, •Epau ` bv, wq “ aEpu, wq ` bEpv, wq for u, v, w P DpEq and a, b P R, •Epu, uq ě 0 for u P DpEq. Given a non-negative definite symmetric bilinear form E on H and α ą 0, define another non-negative definite symmetric bilinear form Eα on H with domain DpEαq :“ DpEq by Eαpu, vq :“ Epu, vq ` αpu, vq, u, v P DpEq. Note that the space DpEq is a pre-Hilbert space with inner product Eα, and Eα and Eβ determine equivalent metrics on DpEq for different α, β ą 0. If DpEq is complete with respect to this metric, then E is said to be closed. In this case, DpEq is then a real Hilbert space with inner product Eα for each α ą 0. A.2 Dirichlet Forms Now consider a σ-finite measure space pX, B, mq and take H to be the Hilbert space L2pX, mq with the usual inner product 164 A Summary of Dirichlet Form Theory pu, vq :“ upxqvpxq mpdxq, u, v P L2pX, mq. żX Call a non-negative definite symmetric bilinear form E on L2pX, mq Markovian if for each ε ą 0, there exists a real function φε : R Ñ R, such that φεptq “ t, t P r0, 1s, ´ ε ď φεptq ď 1 ` ε, t P R, 0 ď φεptq ´ φεpsq ď t ´ s, ´8 ă s ă t ă 8, and when u belongs to DpEq, φε ˝ u also belongs to DpEq with Epφε ˝ u, φε ˝ uq ď Epu, uq. A Dirichlet form is a non-negative definite symmetric bilinear form on L2pX, mq that is Markovian and closed. A non-negative definite symmetric bilinear form E on L2pX, mq is certainly Markovian if whenever u belongs to DpEq, then v “ p0 _ uq ^ 1 also belongs to DpEq and Epv, vq ď Epu, uq. In this case say that the unit contraction acts on E. It turns out the if the form is closed, then the form is Markovian if and only if the unit contraction acts on it. Similarly, say that a function v is called a normal contraction of a function u if |vpxq ´ vpyq| ď |upxq ´ upyq|, x, y P X, |vpxq| ď |upxq|, x P X, and say that v P L2pX, mq a normal contraction of u P L2pX, mq if some Borel version of v is a normal contraction of some Borel version of u. Say that normal contractions act on E if whenever v is a normal contraction of u P DpEq, then v P DpEq and Epv, vq ď Epu, uq. It also turns out that if the form is closed, then the form is Markovian if and only if the unit contraction acts on it. Example A.1. Let X Ď R be an open subinterval and suppose that m is a Radon measures on X with support all of X. Define a non-negative definite symmetric bilinear form by 1 dupxq dvpxq Epu, vq :“ dx 2 dx dx żX on the domain DpEq :“ tu P L2pX, mq : u is absolutely continuous and Epu, uq ă 8u. We claim that E is a Dirichlet form on L2pX, mq. A.2 Dirichlet Forms 165 It is easy to check that the unit contraction acts on E. To show the form is closed, take any E1-Cauchy sequence tu`u. Then tdu`{dxu converges to some 2 2 2 f P L pX, dxq in L pX, dxq. Also, tu`u converges to some u P L pX, mq in L2pX, mq. From this and the inequality |upaq ´ upbq|2 ď 2|a ´ b|Epu, uq, a, b P X, we conclude that there is a subsequence t`ku such that u`k converges to a continuous functionu ˜ uniformly on each bounded closed subinterval of X. Obviouslyu ˜ “ u m-a.e. For all infinitely differentiable compactly supported functions φ on X, an integration by parts shows that du pxq fpxqφpxq dx “ lim `k φpxq dx `kÑ8 dx żX żX 1 1 “ ´ lim u`k pxqφ pxq dx “ ´ u˜pxqφ pxq dx. `kÑ8 żX żX This implies thatu ˜ is absolutely continuous and du˜{dx “ f. Hence,u ˜ P DpEq and tu`u is E1-convergent tou ˜. Example A.2. Consider a locally compact metric space pX, ρq equipped with a Radon measure m. Suppose that we are given a kernel j on X ˆ BpXq satisfying the following conditions. • For any ε ą 0, jpx, XzBεpxqq is, as a function of x P X, locally integrable with respect to m. Here, as usual, Bεpxq is the ball around x of radius ε. • X upxq pjvqpxq mpdxq “ X pjuqpxq vpxq mpdxq for all u, v P pBpXq. Then,ş j determines a symmetricş Radon measure J on X ˆ Xz∆, where ∆ is the diagonal, by fpx, yq Jpdx, dyq :“ fpx, yq jpx, dyq mpdxq. żXˆXz∆ żX "żX * Put Epu, vq :“ pupxq ´ upyqqpvpxq ´ vpyqq Jpdx, dyq żXˆXz∆ on the domain DpEq :“ tu P L2pX, mq : Epu, uq ă 8u. We claim that E is a Dirichlet form on L2pX, mq provided that DpEq is dense in L2pX, mq. It is clear that E is non-negative definite, symmetric, and bilinear. We next show that for a Borel function u that u “ 0 m-a.e. implies that Epu, uq “ 0. Suppose that u “ 0 m-a.e. Put ΓK,ε “ tpx, yq P K ˆ K : ρpx, yq ą εu for ε ą 0 and K compact. Then 166 A Summary of Dirichlet Form Theory pupxq ´ upyqq2 Jpdx, dyq ď 2 pupxq2 ` upyq2q Jpdx, dyq żΓK,ε żΓK,ε 2 2 “ 4 upxq Jpdx, dyq ď 4 upxq jpx, XzBεpxqq mpdxq “ 0. żΓK,ε żK Letting ε Ó 0 and K Ò X gives Epu, uq “ 0. It is clear that every normal contraction operates on the form and so the form is Markovian. To prove that the form is closed, consider a sequence tu`u in DpEq such that lim`,mÑ8 E1pu` ´ um, u` ´ umq Ñ 0. Since tu`u converges 2 in L pX, mq, there is a subsequence t`ku and a set N P BpXq with mpNq “ 0 such that tu`k pxqu converges on XzN. Putu ˜`k pxq “ ulk pxq on XzN and u˜`k pxq “ 0 on N. Thenu ˜`k pxq has a limit upxq everywhere and u` converges to u in L2pX, mq. Moreover, Epu ´ um, u ´ umq 2 “ lim tpu`k pxq ´ u`k pyqq ´ pumpxq ´ umpyqqu Jpdx, dyq `kÑ8 żXˆXz∆ ď lim inf Epulk ´ um, ulk ´ umq. lkÑ8 The last term can be made arbitrarily small for sufficiently large m. Thus, um is E1´convergent to u P DpEq, as required. A.3 Semigroups and Resolvents Suppose again that we have a real Hilbert space H with inner product p¨, ¨q. Consider a family tTtutą0 of linear operators on H satisfying the following conditions: • each Tt is a self-adjoint operator with domain H, • TsTt “ Ts`t, s, t ą 0 (that is, tTtutą0 is a semigroup), • pTtu, Ttuq ď pu, uq, t ą 0, u P H (that is, each Tt is a contraction). We say that tTtutą0 is strongly continuous if, in addition, • limtÓ0pTtu ´ u, Ttu ´ uq “ 0 for all u P H. A resolvent on H is a family tGαuαą0 of linear operators on H satisfying the following conditions: • Gα is a self-adjoint operator with domain H, • Gα ´ Gβ ` pα ´ βqGαGβ “ 0 (the resolvent equation), • each operator αGα is a contraction. The resolvent is said to be strongly continuous if, in addition, • limαÑ8pαGαu ´ u, αGαu ´ uq “ 0 for all u P H. A.5 Spectral Theory 167 Example A.3. Given a strongly continuous semigroup tTtutą0 on H, the family of operators 8 ´αt Gαu :“ e Ttu dt ż0 is a strongly continuous resolvent on H called the resolvent of the given semi- group. The semigroup may be recovered from the resolvent via the Yosida approximation 8 n ´tβ ptβq n Ttu “ lim e pβGβq u, u P H. βÑ8 n! n“0 ÿ A.4 Generators The generator A of a strongly continuous semigroup tTtutą0 on H is defined by T u ´ u Au :“ lim t tÓ0 t on the domain DpAq consisting of those u P H such that the limit exists. Suppose that tGαuαą0 is a strongly continuous resolvent on H. Note that if Gαu “ 0, then, by the resolvent equation, Gβu “ 0 for all β ą 0, and, by strong continuity, u “ limβÑ8 βGβu “ 0. Thus, the operator Gα is invertible and we can set ´1 Au :“ αu ´ Gα u on the domain DpAq :“ GαpHq. This operator A is easily seen to be indepen- dent of α ą 0 and is called the generator of the resolvent. tGαuαą0. Lemma A.4. The generator of a strongly continuous semigroup on H coin- cides with the generator of its resolvent, and the generator is a non-positive definite self-adjoint operator. A.5 Spectral Theory A self-adjoint operator S on H with domain H satisfying S2 “ S is called a projection.
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