Thermal potentials on a Hilbert space
type: Item
instance of: scholarly article
title: Thermal potentials on a Hilbert space
zbMATH Open document ID: 0585.46037
DOI: 10.1007/BF01139989
author: N. V. Norin
published in: Mathematical Notes
publication date: 1984-01-01T00:00:00Z
review text: Let (\(\Phi\) ',(.,.)) be a separable Hilbert space and let \(A: \Phi\) '\(\to \Phi '\) be a self-adjoint positive Hilbert-Schmidt operator. Consider the space \(H=A\Phi '\) and \(\Phi =A^ 2\Phi '\) endowed with the inner products \((.,.)_ H=(A^{-1}.,A^{-1}.)\) and \((.,.)_{\Phi}=(A^{- 2}.,A^{-2}.)\), respectively. The central concepts of this paper are the double-layer potential \(v: \Phi\) '\(\times [0,T]\to R\) defined by \[ v(x,y)=\int^{t}_{0}\int_{S}\rho (\xi,\tau)p^ S_{2(t- \tau)}(x,d\xi)d\tau \] and the single-layer potential \(u: \Phi\) \(\times [0,T]\to R\) defined by \[ u(x,y)=-\int^{t}_{0}\int_{S}\rho (\xi,\tau)\frac{(\eta (\xi),\xi -x)}{2(t-x)}\quad p^ S_{2(t- \tau)}(x,d\xi)d\tau \] where \(S=\{\xi \in \Phi ';\| \xi \| =1\}\), \(\{p_ t;t>0\}\) is the family of Gaussian measures on \(\Phi\) ' and \(\rho\) : \(S\times [0,T]\to R\) is a bounded function. The author proves under certain natural hypothesis that the two potentials are Fréchet infinitely differentiable on H in each point (x,t) with \(x\not\in S\), \(t\in [0,T]\). Also, explicit formulas for differentials are obtained. If \(\rho \in C_ b(S\times [0,T])\), \(x_ 0\in S\) and \(x\to x_ 0\), \(x\not\in S\), it is proved that \(d_{\bar n(x_ 0)}v(x,t)\to 1/2\rho (x_ 0,t)+d_{\bar n(x_ 0)}v(x_ 0,t)\), u(x,t)\(\to -1/2\rho (x_ 0,t)+u(x_ 0,t)\). Using a Wiener process on \(\Phi\) ' the author establishes with the aid of heat potential, sufficient conditions for the existence of solutions of generalized heat equation.
Mathematics Subject Classification ID: 46F25
Mathematics Subject Classification ID: 35K05
Mathematics Subject Classification ID: 46G12
Mathematics Subject Classification ID: 60J45
Mathematics Subject Classification ID: 28C20
Mathematics Subject Classification ID: 35C10
Mathematics Subject Classification ID: 31A35
zbMATH DE Number: 3937819
zbMATH Keywords: self-adjoint positive Hilbert-Schmidt operator
zbMATH Keywords: double-layer potential
zbMATH Keywords: single-layer potential
zbMATH Keywords: Gaussian measures
zbMATH Keywords: Fréchet infinitely differentiable
zbMATH Keywords: explicit formulas for differentials
zbMATH Keywords: Wiener process
zbMATH Keywords: heat potential
zbMATH Keywords: existence of solutions of generalized heat equation
MaRDI profile type: Publication
cites work: Q5642761
cites work: The theory of differentiation in linear topological spaces
cites work: Gaussian measures in Banach spaces
cites work: Smooth functions on Banach spaces
cites work: Potential theory on Hilbert space
cites work: Q5628553
cites work: Q5606327
full work available at URL: https://doi.org/10.1007/bf01139989
OpenAlex ID: W1964695842
description: scientific article; zbMATH DE number 3937819