Brownian Motion with Darning applied to KL and BF equations for - PowerPoint PPT Presentation

Brownian Motion with Darning applied to KL and BF equations for planar slit domains Masatoshi Fukushima with Z.-Q. Chen and S. Rohde September 26, 2012 at Okayama University Stochastic Analysis and Applications German Japanese bilateral project September 26, 2012 at Okayama University Stochastic Masatoshi Fukushima with Z.-Q. Chen and S. Rohde () BMD applied to KL and BF equations / 30

Introduction 1 Objective of my talk Known facts in simply connected case ( N = 0) BMD and its complex Poisson kernel 2 Definition of BMD Complex Poisson kernel of BMD Solving continuity and differentiability problems 3 Strategy Probabilistic expression of ℑ g t ( z ) and continuity of g t ( z ) Lipschitz continuity of BMD-complex Poisson kernel Bauer-Friedrich equation of slit motions 4 BF and KL equations for a given ξ ( t ) 5 Solving BF equation for a given continuous ξ ( t ) Solving KL equation for a given continuous ξ ( t ) Basic properties of ( s ( t ) , ξ ( t )) for a given random curves γ 6 September 26, 2012 at Okayama University Stochastic Masatoshi Fukushima with Z.-Q. Chen and S. Rohde () BMD applied to KL and BF equations / 30

Introduction Objective of my talk Objective of my talk A domain of the form D = H \ ∪ N k =1 C k is called a standard slit domain, where H is the upper half-plane and { C k } are mutually disjoint line segments parallel to x -axis contained in H . We fix a standard slit domain D and consider a Jordan arc γ : [0 , t γ ] → D , γ (0) ∈ ∂ H , γ (0 , t γ ] ⊂ D . (1.1) For each t ∈ [0 , t γ ], let g t : D \ γ [0 , t ] → D t (1.2) be the unique conformal map from D \ γ [0 , t ] onto a standard slit domain D t = H \ ∪ N k =1 C k ( t ) satisfying a hydrodynamic normalization g t ( z ) = z + a t z → ∞ . z + o (1) , (1.3) a t is called half-plane capacity and it can be shown to be a strictly increasing left-continuous function of t with a 0 = 0 . September 26, 2012 at Okayama University Stochastic Masatoshi Fukushima with Z.-Q. Chen and S. Rohde () BMD applied to KL and BF equations / 30

Introduction Objective of my talk We write ξ ( t ) = g t ( γ ( t )) ( ∈ ∂ H ) , 0 ≤ t ≤ t γ . (1.4) In [BF08] On chordal and bilateral SLE in multiply connected domains, Math. Z. 258 (2008), 241-265 R.O. Bauer and R.M. Friedrich have derived a chordal Komatu-Loewner equation ∂ − g t ( z ) = − π Ψ t ( g t ( z ) , ξ ( t )) , g 0 ( z ) = z ∈ D , 0 < t ≤ t γ , (1.5) ∂ a t where ∂ − g t ( z ) denotes the left partial derivative with respect to a t . ∂ a t This is an extension of the radial Komatu-Loewner equation obtained first by Y. Komatu [K50] On conformal slit mapping of multiply-conected domains, Proc. Japan Acad. 26 (1950), 26-31 and later by Bauer-Friedrich [BF06] On radial stochastic Loewner evolution in multiply connected domains, J. Funct. Anal. 237 (2006), 565-588 September 26, 2012 at Okayama University Stochastic Masatoshi Fukushima with Z.-Q. Chen and S. Rohde () BMD applied to KL and BF equations / 30

Introduction Objective of my talk The kernel Ψ t ( z , ζ ) , z ∈ D t , ζ ∈ ∂ H , appearing in (1.5) is an analytic function of z ∈ D t whose imaginary part is constant on each slit C k ( t ) of the domain D t . It is explicitly expressed in terms of the classical Green function of the domain D t . However the following problems have not been solved neither in the radial case [K50], [BF06] nor in the chordal case [BF08]: Problem 1 (continuity). Is a t continuous in t ? Problem 2 (differentiablility). If a t were continuous in t , the curve γ can be reparametrized in a way that a t = 2 t , 0 ≤ t ≤ t γ . Is g t ( z ) differentiable in t so that (1.5) can be converted to the genuine KL equation ? d dt g t ( z ) = − 2 π Ψ t ( g t ( z ) , ξ ( t )) , g 0 ( z ) = z ∈ D , 0 < t ≤ t γ . (1.6) September 26, 2012 at Okayama University Stochastic Masatoshi Fukushima with Z.-Q. Chen and S. Rohde () BMD applied to KL and BF equations / 30

Introduction Objective of my talk g t can be extended as a homeomorphism between ∂ ( D \ γ [0 , t ]) and ∂ D t . The slit C k is homeomorphic with the image slit C k ( t ) by g t for each 1 ≤ k ≤ N . Denote by z k ( t ) , z ′ k ( t ) the left and right endpoints of C k ( t ) . [BF06], [BF08] went on further to make the following claims: Claim 1. The endpoints are subjected to the Bauer-Friedrch equation d d dt z ′ k ( t ) = − 2 π Ψ t ( z ′ dt z k ( t ) = − 2 π Ψ t ( z k ( t ) , ξ ( t )) , k ( t ) , ξ ( t )) , (1.7) Claim 2. Conversely, given a continuous function ξ ( t ) on the boundary ∂ H , the BF-equation (1.7) can be solved uniquely in z k ( t ) , z ′ k ( t ), and then the KL-equation (1.6) can be solved uniquely in g t ( z ) . We aim at answering Problems 1 and 2 affirmatively, establishing the genuine KL-equation (1.6) with Ψ t ( z , ζ ) being the complex Poisson kernel of BMD on D t and moreover legitimating Claims 1 and 2 made by Bauer-Friedrich. September 26, 2012 at Okayama University Stochastic Masatoshi Fukushima with Z.-Q. Chen and S. Rohde () BMD applied to KL and BF equations / 30

Introduction Known facts in simply connected case ( N = 0) Known facts in simply connected case ( N = 0) The continuity of a t follows easily from the Carath´ eodory convegence theorem. The continuity of ξ ( t ) ∈ ∂ H can also be shown by an complex analyitc argument. D t = H and the complex Poisson kernel of H is given by Ψ t ( z , ζ ) = Ψ( z , ζ ) = − 1 1 ( z − ζ ) π with ℑ Ψ( z , ζ ) = 1 1 ( x − ζ ) 2 + y 2 , z = x + iy , π being the Poisson kernel of ABM on H The equation (1.5) is reduced to the well known Loewner equation ∂ 2 ∂ t g t ( z ) = , g 0 ( z ) = z , (1.8) g t ( z ) − ξ t under the reparametrization a t = 2 t . Given a continuous motion ξ ( t ) on ∂ H , { g t } and γ can be recovered by solving the Loewner equation (1.8). September 26, 2012 at Okayama University Stochastic Masatoshi Fukushima with Z.-Q. Chen and S. Rohde () BMD applied to KL and BF equations / 30

Introduction Known facts in simply connected case ( N = 0) Given a probability measure on the collection of continuous curves γ on H connecting 0 and ∞ that satisfies a domain Markov property and conformal invariance, the associated random motion ξ ( t ) equals √ κ B t for κ > 0 and the Brownian motion B t . Conversely, given ξ ( t ) = √ κ B t on ∂ H , the associated trace γ of the stochastic(Schramm) Loewner evolution (SLE) { g t } behaves differently according to the parameter κ and is linked to scaling limits of certain random processes. September 26, 2012 at Okayama University Stochastic Masatoshi Fukushima with Z.-Q. Chen and S. Rohde () BMD applied to KL and BF equations / 30

BMD and its complex Poisson kernel Definition of BMD Definition of BMD Let D = H \ ∪ N k =1 C k be a standard slit domain. A Brownian motion with darning (BMD) Z ∗ for D is, roughly speaking, a diffusion process on H absorbed at ∂ H and reflected at each slit C j but by regarding C j as a single point c ∗ k . To be more precise, let D ∗ = D ∪ K ∗ , K ∗ = { c ∗ 1 , c ∗ 2 , · · · , c ∗ N } (2.1) j of each point c j ∗ in D ∗ by { c ∗ and define a negihborhood U ∗ j } ∪ ( U j \ C j ) for any neighborhood U j of C j in H . Denote by m the Lebesgue measure on D and by m ∗ its zero extension to D ∗ . Let Z 0 = ( Z 0 t , P 0 z ) be the absorbing Brownian motion(ABM) on D . In [CF] Z.-Q. Chen and M. Fukushima, Symmetric Markov Processes, Time Changes, and Boundary Theory , Princeton University Press, 2012, the BMD Z ∗ for D is characterized as a unique m ∗ -symmetric diffusion extension of Z 0 from D to D ∗ with no killing at K ∗ . September 26, 2012 at Okayama University Stochastic Masatoshi Fukushima with Z.-Q. Chen and S. Rohde () BMD applied to KL and BF equations / 30

BMD and its complex Poisson kernel Definition of BMD Let ( E ∗ , F ∗ ) be the Dirichlet form of Z ∗ on L 2 ( D ∗ ; m ∗ ) = L 2 ( D ; m ) . It is a regular strongly local Dirichlet form on L 2 ( D ∗ ; m ∗ ) described as { E ∗ ( u , v ) = 1 u , v ∈ F ∗ , 2 D ( u , v ) , F ∗ = { u ∈ W 1 , 2 ( H ) : u is constant D -q.e on each C j } . 0 The uniquely associated diffusion on D ∗ is the BMD Z ∗ . Let γ be an analytic Jordan curve surrounding a slit C j , namely, γ ⊂ D , ins γ ⊃ C j , ins γ ∩ C k = ∅ , k � = j . For a harmonic function u defined in a neighborhood of C j , the value ∫ ∂ u ( ζ ) ds ( ζ ) ∂ n ζ γ is independent of the choice of such curve γ with the normal vector n pointing toward C j and arc length s . This value is called the period of u around C j . September 26, 2012 at Okayama University Stochastic Masatoshi Fukushima with Z.-Q. Chen and S. Rohde () BMD applied to KL and BF equations / 30