Skin moisturizing has drawn attention in terms of beauty and anti-aging industries. However, it is difficult to observe the inside of the epidermis and the relationship between the epidermis and water content is not yet clear. Computational simulations can be useful in understanding such mechanisms of skin formation. A particle model was used to simulate three-dimensional skin turnover, and the results reproduced the epidermal skin turnover phenomenon. In this study, a diffusion model is introduced into this simulation model and a moisture diffusion analysis of the epidermis was performed. In particular, transepidermal water loss (TEWL) was modeled by considering diffusion and surface evaporation in the stratum corneum and other layers. The relationship between the moisture content and the keratin detachment was considered, and the exfoliation condition of keratin based on the moisture content was calculated in the model. As a result, it was possible to calculate the intraepidermal water content distribution in the skin using the particle model. It was also possible to reproduce phenomena such as keratin thickening due to increase of TEWL. This phenomenon is consistent with cases of dry skin. In the future, it will be necessary to introduce a change in TEWL according to the thickness of the stratum corneum and the diffusion coefficient.

Abstract:
We prove that the martingale dimensions for canonical diffusion processes on a class of self-similar sets including nested fractals are always one. This provides an affirmative answer to the conjecture of S. Kusuoka [Publ. Res. Inst. Math. Sci. 25 (1989) 659--680].

Abstract:
In Euclidean space, the integration by parts formula for a set of finite perimeter is expressed by the integration with respect to a type of surface measure. According to geometric measure theory, this surface measure is realized by the one-codimensional Hausdorff measure restricted on the reduced boundary and/or the measure-theoretic boundary, which may be strictly smaller than the topological boundary. In this paper, we discuss the counterpart of this measure in the abstract Wiener space, which is a typical infinite-dimensional space. We introduce the concept of the measure-theoretic boundary in the Wiener space and provide the integration by parts formula for sets of finite perimeter. The formula is presented in terms of the integration with respect to the one-codimensional Hausdorff-Gauss measure restricted on the measure-theoretic boundary.

Abstract:
We introduce the concept of index for regular Dirichlet forms by means of energy measures, and discuss its properties. In particular, it is proved that the index of strong local regular Dirichlet forms is identical with the martingale dimension of the associated diffusion processes. As an application, a class of self-similar fractals is taken up as an underlying space. We prove that first-order derivatives can be defined for functions in the domain of the Dirichlet forms and their total energies are represented as the square integrals of the derivatives.

Abstract:
We introduce Riemannian-like structures associated with strong local Dirichlet forms on general state spaces. Such structures justify the principle that the pointwise index of the Dirichlet form represents the effective dimension of the virtual tangent space at each point. The concept of differentiations of functions is studied, and an application to stochastic analysis is presented.

Abstract:
Given strong local Dirichlet forms and $\mathbb{R}^N$-valued functions on a metrizable space, we introduce the concepts of geodesic distance and intrinsic distance on the basis of these objects. They are defined in a geometric and an analytic way, respectively, and they are closely related with each other in some classical situations. In this paper, we study the relations of these distances when the underlying space has a fractal structure. In particular, we prove their coincidence for a class of self-similar fractals.

Abstract:
We confirm, in a more general framework, a part of the conjecture posed by R. Bell, C.-W. Ho, and R. S. Strichartz [Energy measures of harmonic functions on the Sierpi\'nski gasket, Indiana Univ. Math. J. 63 (2014), 831--868] on the distribution of energy measures for the canonical Dirichlet form on the two-dimensional standard Sierpinski gasket.

Abstract:
We study upper estimates of the martingale dimension $d_m$ of diffusion processes associated with strong local Dirichlet forms. By applying a general strategy to self-similar Dirichlet forms on self-similar fractals, we prove that $d_m=1$ for natural diffusions on post-critically finite self-similar sets and that $d_m$ is dominated by the spectral dimension for the Brownian motion on Sierpinski carpets.

Abstract:
We introduce the concept of functions of locally bounded variation on abstract Wiener spaces and study their properties. Some nontrivial examples and applications to stochastic analysis are also discussed.

Abstract:
We consider the $(1,2)$-Sobolev space $W^{1,2}(U)$ on subsets $U$ in an abstract Wiener space, which is regarded as a canonical Dirichlet space on $U$. We prove that $W^{1,2}(U)$ has smooth cylindrical functions as a dense subset if $U$ is $H$-convex and $H$-open. For the proof, the relations between $H$-notions and quasi-notions are also studied.