Generalizations to different state spaces

1. ‘state space’ D ⊂ R d is a bounded open convex set, D is its closure and ∂ D = D\D. 2. ‘fixed shape’ g ∗ : D → R is the unique continuous solution of − 1 2 g ∗ = 1 on D g ∗ = 0 on ∂ D, 2.2.5 with = d i =1 ∂ 2 ∂ x i 2 the Laplacian. 3. ‘diffusion function’ H is the class of functions g : D → [0, ∞ satisfying i g ≤ Mg ∗ for some M ∞ ii g 0 on D iii g continuous on D. 2.2.6 4. ‘attraction point’ θ ∈ D. 5. ‘attraction constant’ c ∈ 0, ∞. With these ingredients we let our basic diffusion equation be the SDE: d X t = cθ − X t dt + 2gX t d B t , 2.2.7 where B t t ≥0 is standard d-dimensional Brownian motion. Solutions of 2.2.7 solve the martingale problem for the operator A with domain D A given by A f x : = cθ − x · ∇ + gx f x D A : = C 2 D, 2.2.8 where ∇ = ∂ ∂ x i , . . . , ∂ ∂ x d and · denotes inner product. The martingale problem for A is well-posed if and only if, for each initial condition on D, the SDE 2.2.7 has a unique weak solution X t t ≥0 . In this case, the operator A has a unique extension to a generator of a Feller semigroup, and X t t ≥0 is the associated Feller process. 4 By a continuous probability kernel on D we mean a continuous map K : D → P D, written x → K x , where P D is the space of probability mea- sures on D, equipped with the topology of weak convergence. We equip the space K D : = C D, P D of probability kernels on D with the topology of uniform convergence. Since P D is compact and Hausdorff, there is a unique uniform structure defining the topology, and we can unambiguously speak about uniform 4 For a discussion of these facts, see the footnote at Theorem 2.1.1. convergence of P D-valued functions. There exists a natural identification be- tween continuous probability kernels K ∈ K D and continuous positive linear operators K : C D → C D satisfying K 1 = 1, the correspondence being given by K f x = D K x d y f y f ∈ C D. 2.2.9 In this identification, the composition of two kernels is given by K L x d y = D K x d yL y d z. 2.2.10 The convergence of operators K n → K in the topology on K D is equivalent to the convergence of the functions K n f → K f , uniformly on D for all f ∈ C D. In order to be able to define our renormalization transformation, we introduce a new class H ′ of diffusion functions as follows: 3 ′ . H ′ is the class of all functions g ∈ H such that for all c ∈ 0, ∞ and θ ∈ D: 1 The martingale problem associated with the operator A in 2.2.8 is well- posed. 2 The diffusion associated with 2.2.7 has a unique equilibrium ν g,c θ . Here, by an equilibrium we mean a stationary distribution of 2.2.7. As we shall see in section 2.2.4, these assumptions are satisfied for many g ∈ H . It turns out that the map θ → ν g,c θ is continuous, and so the equilibrium of 2.2.7 is a continuous probability kernel on D as a function of the parameter θ . Theorem 2.2.1 For each g ∈ H ′ and c ∈ 0, ∞ there exists a continuous proba- bility kernel ν g,c ∈ K D such that, for each θ ∈ D, ν g,c θ is the equilibrium of the diffusion in 2.2.7. For g ∈ H ′ and c ∈ 0, ∞, we now define our ‘renormalization transformation’ as F c gθ : = ν g,c gθ = D gxν g,c θ d x. 2.2.11 In order to speak about the iterates of F c , we need a subclass of H ′ that is closed under the F c ’s. For this we may take the largest such subclass, so we define one more class of diffusion functions: 3 ′′ . H ′′ is the union of all G ⊂ H ′ such that F c G ⊂ G for all c ∈ 0, ∞. With these definitions, we have the following result. Theorem 2.2.2 For all c ∈ 0, ∞: F c H ′ ⊂ H . It is at present not known if H = H ′ , but Theorem 2.2.2 implies at least that if H ′ = H , then H ′′ = H . The next result generalizes Theorem 2.1.7 recall the composition of probabil- ity kernels defined in 2.2.10: Theorem 2.2.3 For g ∈ H ′′ and k ≥ 1, let K g,k be given by K g,k : = ν F k −1 g,c k · · · ν g,c 1 , 2.2.12 where F k g : = F c k ◦ · · · ◦ F c 1 g is the k-th iterate of the renormalization transfor- mations F c applied to g F g = g. If k c −1 k = ∞, then in the sense of uniform convergence of probability kernels: K g,k → K ∞ as k → ∞, 2.2.13 where the limiting kernel K ∞ is universal in g and given by K ∞ θ d x = P[B θ τ ∈ dx], 2.2.14 where B θ t t ≥0 is Brownian motion starting in θ and τ : = inf{t ≥ 0 : B θ t ∈ ∂ D}. The following generalizes Theorem 2.1.9: Theorem 2.2.4 a Let g ∗ be as in 2.2.5. If g ∗ ∈ H ′ , then rg ∗ ∈ H ′′ for all r 0. Moreover, the 1-parameter family of functions rg ∗ r 0 are fixed shapes under F c : F c rg ∗ = c c + r rg ∗ . 2.2.15 b If k c −1 k = ∞, then for all g ∈ H ′′ lim k →∞ σ k F k g = g ∗ uniformly on D, 2.2.16 where σ k : = k l =1 c −1 l . c If, in addition to the assumptions in b, there exists a λ 0 such that g ≥ λg ∗ , then lim k →∞ σ k F k g − g ∗ H = 0, 2.2.17 where the norm · H is given by g H : = sup x ∈D gx g ∗ x . 2.2.18 In d = 1, formula 2.2.17 is in fact known to hold under somewhat weaker condi- tions on g see Theorem 2.1.9 c.