Dirichlet forms for singular diffusion in higher dimensions

Abstract

We describe singular diffusion in bounded subsets of R n by form methods and characterize the associated operator. We also prove positivity and contractivity of the corresponding semigroup. This results in a description of a stochastic process moving according to classical diffusion in one part of , where jumps are allowed through the rest of .

On the construction and convergence of traces of forms

“…For an application of the situation in Lemma 3.4 see [FS15,SV11]. By means of Proposition 3.3 we can now handle the following case.…”

“…In [Pos16], Post used so-called boundary pairs (referring to the case that J has a dense kernel) to construct a family of operators related to the associated operator to the trace form. Moreover, there are applications in the context of Dirichlet forms and singular diffusions, see [SV11,FS15].…”

“…This includes the square root of the Laplacian as obtained in [CS07] revisited in the context of forms, but also traces on (maybe small) subsets, wich can correspond to singular diffusions; cf. [SV11,FS15]. Starting from Section 5 we focus on Dirichlet forms.…”

An approximation of solutions to heat equations defined by generalized measure theoretic Laplacians

“…This definition involves the derivative with respect to μ. If a function f has a representation given by f (x) = This operator has been widely studied, for example with an emphasis on addressing questions of the spectral asymptotics and further analytical properties [2,3,11-20, 22,23,33,34,36,37], diffusion processes [27,30,31], wave equations [5] and higherdimensional generalizations [21,35,39].…”

“…Let A be the generator of a strongly continuous semigroup (S t ) t≥0 on a Banach space X . Then, for each f ∈ D(A) the abstract heat equation∂u ∂t (t) = Au(t), t ≥ 0 u(0) = f (21)has a unique classical solution on X given byu : [0, ∞) → X, t → S t f,meaning that u is continuously differentiable with respect to X , u(t) ∈ D (A) and(21) holds for all t ≥ 0.Let T > 0 and f ∈ D ¯ b μ . Theorem 1.2 implies that the classical solution to∂u n ∂t (t) =¯ b μ n u n (t), u n (0) = π n f converges uniformly for (t, x) ∈ [0, T ] × [0, 1] to the classical solution to ∂u ∂t (t) =¯ b μ u(t), u(0) = fas n → ∞, assuming that π n f ∈ D ¯ b μ n .…”