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Diffusion Processes with Reflection

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Diffusion Processes with Reflection Diffusion Processes with Reflection vorgelegt von Dipl. math. oec. Sebastian Andres aus Berlin Von der Fakult¨at II – Mathematik und Naturwissenschaften der Technischen Universit¨at Berlin zur Erlangung des akademischen Grades Doktor der Naturwissenschaften – Dr. rer. nat. – genehmigte Dissertation Promotionsausschuss Vorsitzender: Prof. Dr. Reinhold Schneider Gutachter: Prof. Dr. Jean-Dominique Deuschel Prof. Dr. Lorenzo Zambotti Tag der wissenschaftlichen Aussprache: 13. Mai 2009 Berlin 2009 D 83 Preface Es gibt nichts Praktischeres als eine gute Theorie. Immanuel Kant In modern probability theory diffusion processes with reflection arise in various manners. As an example let us consider the reflected Brownian motion, that is a Brownian motion on the positive real line with reflection in zero. There are several possibilities to construct this process, the simplest and most intuitive one is certainly the following: Given a standard Brownian motion B, starting in some x 0, a reflected Brownian motion is obtained by taking the absolute value ≥ B . By the Itˆo-Tanaka formula we have | | t B = x + sign(B ) dB + L , | t| s s t Z0 where (Lt)t 0 is the local time of B in zero, i.e. it is continuous, nondecreasing and supp(dL) ≥ t ⊆ t 0 : B = 0 . Note that β := sign(X ) dB is again a Brownian motion by L´evy’s { ≥ t } t 0 s s characterization theorem. R Another possibility to construct a reflected Brownian motion is to apply the Skorohod re- flection principle. Given a standard Brownian motion W there exists a unique pair (X,L) such that X = x + W + L , x 0, t 0, (0.1) t t t ≥ ≥ where X 0 for all t, L starts in zero and is continuous, monotone nondecreasing and supp(dL) t ≥ ⊆ t 0: Xt =0 , i.e. it increases only at those times, when X is zero. Moreover, the local time { ≥ } + L is given by Lt =[ x infs t Ws] . Here the reflected Brownian motion (Xt) is constructed by − − ≤ adding the local time L. Equation (0.1) is a simple example for a so called Skorohod-SDE. Since β defined above is a Brownian motion, one can easily check that X =d B . | | A much more abstract and analytic method to introduce the reflected Brownian motion is to define it via its Dirichlet form. The reflected Brownian motion is the diffusion associated with the Dirichlet form 1 ∞ 1,2 (f,g) = f ′(x) g′(x) dx, f,g W ([0, )). E 2 ∈ ∞ Z0 iii iv Another meaningful class of stochastic processes involving some kind of reflection are Bessel processes. The Bessel process ρδ with Bessel dimension δ > 0 is again a diffusion process taking nonnegative values, which is associated with the Dirichlet form δ ∞ δ 1 1,2 δ 1 (f,g) = C f ′(x) g′(x) x − dx, f,g W ([0, ),x − dx), E δ ∈ ∞ Z0 Cδ denoting a positive constant. Usually Bessel processes are introduced by defining the square of a Bessel process (see e.g. Chapter XI in [56]). For integer dimensions, i.e. for δ [1, ) N, ρδ ∈ ∞ ∩ can be obtained by taking the Euclidian norm of a δ-dimensional Brownian motion. In particular, the one-dimensional Bessel process ρ1 is just the reflected Brownian motion discussed above. For δ > 1, the Bessel process ρδ starting in some x 0 can be described as the unique solution of the ≥ SDE δ 1 t 1 ρδ = x + − ds + B . (0.2) t 2 ρδ t Z0 s The drift term appearing in this equation effects a repulsion, which is strong enough to force the process to stay nonnegative, so that no further reflection term is needed. Finally, for 0 < δ < 1 the situation is much more complicated, because the drift term is now attracting and not repulsive. As a consequence, a representation of ρδ in terms of an SDE like (0.1) or (0.2) is not possible in this case, because ρδ is known not to be a semi-martingale if δ < 1 (see e.g. Section 6 in [57]). Nevertheless, ρδ admits a Fukushima decomposition ρδ = B +(δ 1)H , t 0, t t − t ≥ as the sum of a Brownian motion and a zero energy process (δ 1)H, which also produces the − reflection in this case, see [13] for details. In this thesis, diffusion processes will appear in nearly every fashion mentioned so far. The thesis consists of two parts. The first part contains some pathwise differentiability results for Skorohod SDEs. In the second part a particle approximation of the Wasserstein diffusion is established, where the approximating process can be intepreted as a system of interacting Bessel process with small Bessel dimension. The second part is a joint work with Max von Renesse, TU Berlin. Pathwise Differentiability for SDEs with Reflection Consider some closed bounded connected domain G in Rd, d 2, and for any starting point ≥ x G the following SDE of the Skorohod type: ∈ t t X (x) = x + b(X (x)) dr + w + γ(X (x)) dl (x), t 0, t r t r r ≥ Z0 Z0 ∞ Xt(x) G, dlt(x) 0, 1lG ∂G(Xt(x)) dlt(x)=0, t 0, ∈ ≥ \ ≥ Z0 where w is a d-dimensional Brownian motion and b is a continuously differentiable drift. Fur- thermore, for every x ∂G, γ(x) denotes the direction of reflection at x, for instance in the case ∈ v of normal reflection γ(x) is the inner normal field on ∂G. Finally, l(x) the local time of X(x) in ∂G, i.e. it increases only at those times, when X(x) is at ∂G. In the first part of this thesis we investigate the question whether the mapping x X (x), 7→ t t > 0, is differentiable almost surely and whether one can characterize the derivatives in order to derive a Bismut formula. Deuschel and Zambotti have solved this problem in [26] for the domain G = [0, )d and several related questions have already been considered in [6]. ∞ In Chaper 1 (cf. [8]), we consider the case where G is a convex polyhedron and where the directions of reflection along each face are constant but possibly oblique. The differentiability is obtained for every time t up to the first time when the process X hits two of the faces simul- taneously. Similarly to the results in [26], the derivatives evolve according to a linear ordinary differential equation, when the process X is in the interior of the domain, and they are projected to the tangent space, when it hits the boundary. In particular, when X reaches the end of an excursion interval, the derivative process jumps in the direction of the corresponding direction of reflection. This evolution becomes rather non-trival due to the complicated structure of the set of times, when the process is at the boundary, which is known to a.s. a closed set of zero Lebesgue measure without isolated points. Chapter 2 (cf. [7]) deals with the case where G is a bounded smooth domain with normal reflection at the boundary. The additional difficulties appearing here are caused on one hand by the presence of curvature and on the other hand by the fact that one cannot apply anymore the technical lemma, on which the proofs in [26] and [8] are based, dealing with the time a Brownian path attains its minimum. As a result we obtain an analogous time evolution for the derivatives as described above. In contrast to the results in [26], this evolution gives not a complete characterization of the derivatives in this case. Therefore, a further SDE-like equation is established, characterizing the derivatives in the coordinates w.r.t. a moving frame. Particle Approximation of the Wasserstein Diffusion The Wasserstein diffusion, recently constructed in [66] by using abstract Dirichlet form methods, is roughly speaking a reversible Markov process, taking values in the set of probability measures P on the unit interval, whose intrinsic metric is given by the L2-Wasserstein distance. In order to improve the intuitive and mathematical understanding of that process, we establish a reversible particle system consisting of N diffusing particles on the unit interval, whose associated empirical measure process converges weakly to the Wasserstein diffusion in the high-density limit, while we have to assume Markov uniqueness for the Dirichlet form, which induces the Wasserstein diffusion (cf. [9]). This approximating particle system can be interpreted as a system of coupled, two-sided Bessel processes with small Bessel dimension. In particular, for large N the drift term, which appears in the SDE describing formally the particle system, is attracting and not repulsive. In analogy to the Bessel process it is not contained in the class of Euclidian semi-martingales. A detailed analysis of the approximating system, in particular Feller properties, will be subject of Chapter 3 (cf. [10]) based on harmonic analysis on weighted Sobolev spaces. A crucial step here is to verify the Muckenhoupt condition for the reversible measure, which will imply the doubling property and a uniform local Poincar´einequality. From this one can derive the Feller property using an abstract result by Sturm [59]. vi To obtain the convergence result one has to prove tightness and in a second step one has to identify the limit. While tightness easily follows by some well-established tightness-criteria, the identification of the limit is more difficult. The proof uses the parametrization of in P terms of right-continuous quantile functions. Then, the convergence of the empirical measure process is equivalent to the convergence of the process of the corresponding quantile functions. This convergence is then obtained by proving that the associated Dirichlet forms converge in the Mosco sense to the limiting Dirichlet form, where we use the generalized version of Mosco convergence with varying base spaces established by Kuwai and Shioya (cf.
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