Thermal potentials on a Hilbert space
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.
self-adjoint positive Hilbert-Schmidt operator
existence of solutions of generalized heat equation
scientific article; zbMATH DE number 3937819