Essential self-adjointness of Dirichlet operators on a path space with Gibbs measures via a SPDE approach

Hiroshi KAWABI^{∗} (Osaka University)

This talk is based on a joint work with Michael Röckner (University of Bielefeld & Purdue University) [8]. In this talk, we discuss on essential self-adjointness of a (pre-)Dirichlet operator on an infinite-volume path space C(ℝ, ℝ^{d}) with a Gibbs measure associated with the (formal) Hamiltonian

H(w) := \frac{1}{2} ∫_{ℝ} |w′(x)|_{ℝ^{d}}^{2}dx + ∫_{ℝ} U(w(x))dx,

where U : ℝ^{d} → ℝ is a potential function. In this talk, we impose the following conditions:

(U1) U ∈ C^{1}(ℝ^{d}, ℝ) and there exists a constant K_{1} ∈ ℝ such that the following holds for any z_{1}, z_{2} ∈ ℝ^{d}:

(∇U(z_{1}) − ∇U(z_{2}), z_{1} − z_{2})_{ℝ^{d}} ≥ −K_{1}|z_{1} − z_{2}|^{2}.

(U2) There exist K_{2} > 0 and p > 0 such that |∇U(z)| ≤ K_{2}(1 + |z|^{p}) holds for any z ∈ ℝ^{d}.

(U3) lim_{|z|→∞}U(z) = ∞.

As examples of U satisfying above conditions, we are interested in a square potential and a double-well potential. Those are, U(z) = a|z|^{2} and U(z) = a(|z|^{4} − |z|^{2}), a > 0, respectively.

Before stating our results, we introduce some notations and objects we will be working with. We define a weight function ρ_{r} ∈ C^{∞}(ℝ, ℝ), r ∈ ℝ, by ρ_{r}(x) := e^{rχ}(x), x ∈ ℝ, where χ ∈ C^{∞}(ℝ, ℝ) is a positive symmetric convex function satisfying χ(x) = |x| for |x| ≥ 1. We fix r > 0 and set E = L_{r}^{2}(ℝ, ℝ^{d}) := L^{2}(ℝ, ℝ^{d} : ρ_{−2r}(x)dx). This is a Hilbert space with the inner product defined by

(X, Y)_{E} := ∫_{ℝ} (X(x), Y (x))_{ℝ^{d}}ρ_{−2r}(x)dx, X, Y ∈ E.

Moreover, we set H := L^{2}(ℝ, ℝ^{d}) and denote by ∥ · ∥_{E} and ∥ · ∥_{H} the corresponding norms of E and H, respectively. We also introduce a suitable subspace of C(ℝ, ℝ^{d}). For functions in C(ℝ, ℝ^{d}), we set

∥w∥_{r,∞} := sup_{x∈ℝ}|w(x)|e^{−rχ(x)} for r ∈ ℝ,

and consider

C := ⋂_{r>0} {w ∈ C(ℝ, ℝ^{d}) | ∥w∥_{r,∞} < ∞}.

Then it becomes a Fréchet space with the system of norms ∥ · ∥_{λ,∞}. We easily see that the inclusion C ⊂ E is dense with respect to the topology of E. In what follows, we denote by P(C(ℝ, ℝ^{d})) the class of all probability measures on the space C(ℝ, ℝ^{d}) and let B_{T} and B_{T,c} be the σ-fields of C(ℝ, ℝ^{d}) generated by {w(x); − T ≤ x ≤ T } and {w(x); x ≤ −T, x ≥ T }, respectively.

∗JSPS Post-Doctral Research Fellowship. e-mail: kawabi@sigmath.es.osaka-u.ac.jp

Now, we introduce a Gibbs measure on the path space C(ℝ, ℝ^{d}). Consider the Schrödinger operator H := −\frac{1}{2}Δ+U on L^{2}(ℝ^{d}, ℝ), where Δ := ∑_{i=1}^{d} ∂^{2}/∂z_{i}^{2} is the d-dimensional Laplacian. Then the condition (U3) assures that H has purely discrete spectrum and a complete set of eigenfunctions. We denote by λ_{0} the lowest eigenvalue and by Ω the corresponding normalized eigenfunction in L^{2}(ℝ^{d}, ℝ). It is called by ground state and it decays exponentially.

Let W_{−T,z_{1};T.z_{2}}, T > 0, z_{1}, z_{2} ∈ ℝ^{d}, be the path measure of Brownian bridge such that W_{−T,z_{1};T.z_{2}}(w(−T) = z_{1}, w(T) = z_{2}) = 1. We define μ(A) for A ∈ B_{T}, T > 0, by

μ(A) := e^{2T λ_{0}} ∫_{ℝ^{d}} ∫_{ℝ^{d}} Ω(z_{1})Ω(z_{2})p(2T, z_{1}, z_{2})
×𝔼^{W_{−T,z_{1};T,z_{2}}} [ exp ( − ∫_{−T}^{T} U(w(x))dx); A]dz_{1}dz_{2},  (1)

where p(t, z_{1}, z_{2}) is the transition probability of standard Brownian motion on ℝ^{d}. Then by the Feynman-Kac formula and the Markov property of Brownian motion, we can see that μ is well-defined as an element of P(C(ℝ, ℝ^{d})) and it satisfies the following DLR-equation for every T ∈ ℕ and μ-a.e. ξ ∈ C(ℝ, ℝ^{d}):

μ(dw|B_{T,c})(ξ) = Z_{T,ξ}^{−1} exp ( − ∫_{−T}^{T} U(w(x))dx)W_{−T,ξ(−T);T,ξ(T)}(dw),  (2)

where Z_{T,ξ} is the normalization constant. (See Proposition 2.7 in Iwata [6] or Section 3.1 in Funaki [4].) Although generally there exist other μ ∈ P(C(ℝ, ℝ^{d})) satisfying DLR-equation (2), in this talk we only consider the Gibbs measure μ which has been constructed in (1).

Here we note that the Gibbs measure μ is supported on C by using the standard moment estimates of Brownian motion. Then by the continuity of the inclusion map of C into E, we can also regard μ ∈ P(E) by identifying it with its image measure under the inclusion map.

By virtue of DLR-equation (2), it holds that C_{0}^{∞}(ℝ, ℝ^{d})-quasi-invariance of the Gibbs measure μ, i.e., μ(· + k) and μ are mutually equivalent and μ(k + dw) = Λ(k, w)μ(dw) holds for every k ∈ C_{0}^{∞}(ℝ, ℝ^{d}). The Radon-Nikodym density Λ(k, w) is given by

Λ(k, w) = exp \{ ∫_{ℝ} (U(w(x)) − U(w(x) + k(x)) − \frac{1}{2}|k′(x)|^{2} + (w(x), Δ_{x}k(x))_{ℝ^{d}})dx\},

where Δ_{x} := d^{2}/dx^{2}. (See Theorem 3.21 in [6] or Lemma 4.1 in Funaki [5].) Moreover, since Ω decays exponentially and μ is shift invariant, we see

∫_{E} ( ∫_{ℝ} |w(x)|^{2m}ρ_{−2r}(x)dx)μ(dw) ≤ \frac{1}{r} ∫_{ℝ^{d}} |z|^{2m}Ω(z)^{2}dz < ∞

holds for any m ∈ ℕ and r > 0.

Now we define the space of smooth cylinder functions. We say a function F : E ⟶ ℝ is in a class FC_{b}^{∞} if there exist n ∈ ℕ, {φ_{1}, ⋯, φ_{n}} ⊂ C_{0}^{∞}(ℝ, ℝ^{d}) and a function f ≡ f(α_{1}, ⋯, α_{n}) ∈ C_{b}^{∞}(ℝ^{n}) such that

F (w) ≡ f(⟨w, φ_{1}⟩, ⋯, ⟨w, φ_{n}⟩), w ∈ E.

Here we use the notation ⟨w, φ⟩ := ∫_{ℝ}(w(x), φ(x))_{ℝ^{d}}dx if the integral is absolutely converging. Since C_{0}^{∞}(ℝ, ℝ^{d}) is dense in E, we can see supp(μ) = E. (See Proposition 2.7 in Albeverio and Röckner [1].) Hence two different functions in FC_{b}^{∞} represents two different μ-classes. Note that FC_{b}^{∞} is dense in L^{2}(μ). For F ∈ FC_{b}^{∞}, we also define the H-Fréchet derivative D_{H}F : E → H by

D_{H}F (w)(x) := ∑_{k=1}^{n} \frac{∂f}{∂α_{k}} (⟨w, φ_{1}⟩, ⋯, ⟨w, φ_{n}⟩)φ_{k}(x), x ∈ ℝ.

We consider a pre-Dirichlet form (E, FC_{b}^{∞}) which is given by

E(F, G) = \frac{1}{2} ∫_{E} (D_{H}F (w), D_{H}G(w))_{H}μ(dw), F, G ∈ FC_{b}^{∞}.

Then by virtue of the C_{0}^{∞}(ℝ, ℝ^{d})-quasi-invariance, we obtain the equality

E(F, G) = ( − L_{0}F, G)_{L^{2}(μ)}, F, G ∈ FC_{b}^{∞},  (3)

where L_{0} is given by

L_{0}F (w) = \frac{1}{2} ∑_{i,j=1}^{n} \frac{∂^{2}f}{∂α_{i}∂α_{j}} (⟨w, φ_{1}⟩, ⋯, ⟨w, φ_{n}⟩)⟨φ_{i}, φ_{j}⟩

+\frac{1}{2} ∑_{i=1}^{n} \frac{∂f}{∂α_{i}} (⟨w, φ_{1}⟩, ⋯, ⟨w, φ_{n}⟩) · \{⟨w, Δ_{x}φ_{i}⟩ − ⟨∇U(w(·)), φ_{i}⟩\}.

This means the operator L_{0} is the pre-Dirichlet operator which is associated with the pre-Dirichlet form (E, FC_{b}^{∞}). In particular, (E, FC_{b}^{∞}) is closable on L^{2}(μ). So we can consider D(E) by the completion of FC_{b}^{∞} with respect to E_{1}^{1/2}-norm and (E, D(E)) is a Dirichlet form.

(3) also implies that the operator L_{0} is symmetric in L^{2}(μ). In many applications, it is an important problem whether one has essential self-adjointness for L_{0}, i.e., self-adjointness of the closure (\overline{L}_{2}, Dom(\overline{L}_{2})) of (L_{0}, FC_{b}^{∞}) in L^{2}(μ). The reason is that in general there are many lower bounded self-adjoint extensions L̃_{2} of L_{0} in L^{2}(μ) which therefore define symmetric strongly continuous semigroups {e^{tL̃_{2}}}_{t≥0} generated by them. In fact, there always exists one such extension called the Freidrichs extension which is the operator corresponding to the Dirichlet form (E, D(E)). If L_{0} is essentially self-adjoint, there is hence only one such semigroup.

The following is the main result of this talk, i.e., we show that our semigroup is not only unique but also represented through the solution of a parabolic SPDE (4) on ℝ.

Theorem 1 The pre-Dirichlet operator (L_{0}, FC_{b}^{∞}) is essentially self-adjoint in L^{2}(μ). Moreover the semigroup \{T_{t}\}_{t≥0} generated by \overline{L}_{2} is represented by

T_{t}F (w) = 𝔼[F (X_{t}^{w})], μ-a.s., t ≥ 0,

where X_{t}^{w} = (X_{t}^{w}(x))_{x∈ℝ} is the solution of the SPDE

\{ \begin{matrix}dX_{t}(x) = \frac{1}{2}\{Δ_{x}X_{t}(x) − ∇U(X_{t}(x))\}dt + dB_{t}(x), x ∈ ℝ, t > 0, \\ X_{0}(x) = w(x), \end{matrix}  (4)

where w ∈ C is an initial data and {B_{t}}_{t≥0} is a H-cylindrical Brownian motion over a probability space (Θ, F, ℙ).

Remark 1 Under the conditions (U1) and (U2), Iwata [7] proved that the SPDE (4) has a (pathwise-)unique solution living in C([0, ∞), C) for an initial data w ∈ C.

As a corollary of this theorem, we can also obtain the Markov uniqueness. See Chapter 1 in Eberle [3] for the proof. Here we say that a Dirichlet form (E, Dom(E)) in L^{2}(μ) is an extension of (L_{0}, FC_{b}^{∞}) if FC_{b}^{∞} ⊂ Dom(E) and E(F, G) = ( − L_{0}F, G)_{L^{2}(μ)} for any F ∈ FC_{b}^{∞} and G ∈ Dom(E).

Theorem 2 The Dirichlet form (E, D(E)) is the unique extension of (L_{0}, FC_{b}^{∞}).

References

[1] S. Albeverio and M. Röckner: New developments in theory and applications of Dirichlet forms, in “Stochastic Processes, Physics and Geometry, Ascona/Locarno, Switzerland, 1988” (S. Albeverio et al., Eds.), pp. 27-76, World Scientific, Singapore, 1990.

[2] G. Da Prato and M. Röckner: Singular dissipative stochastic equations in Hilbert spaces, Probab. Theory Relat. Fields 124 (2002), pp. 261-303.

[3] A. Eberle: Uniqueness and non-uniqueness of singular diffusion operators, Lecture Notes in Mathematics 1718, Springer-Verlag Berlin Heidelberg, 1999.

[4] T. Funaki: Derivation of the hydrodynamical equation for one-dimensional Ginzburg-Landau model, Probab. Theory Relat. Fields 82 (1989), pp. 39-93.

[5] T. Funaki: The reversible measure of multi-dimensional Ginzburg-Landau continuum model, Osaka J. Math.28 (1991), pp. 463-494.

[6] K. Iwata: Reversible measures of a P (ϕ)_{1}-time evolution, in “Prob. Meth. in Math. Phys. : Proceedings of Taniguchi symposium” (K. Itô and N. Ikeda eds.), Kinokuniya, (1985), pp. 195-209.

[7] K. Iwata: An infinite dimensional stochastic differential equation with state space C(ℝ), Probab. Theory Relat. Fields 74 (1987), pp. 141-159.

[8] H. Kawabi and M. Röckner: Essential self-adjointness of Dirichlet operators on a path space with Gibbs measures via a SPDE approach, preprint, 29 pages, 2005.