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We show the uniqueness and the existence of viscosity solutions of Hamilton-Jacobi equations on a smooth Banach space. The tool used is the variational principle of Deville, Godefroy and Zizler. The existence is given by Perron’s method. So we give a comparison assertion for semicontinuous solutions.
Let H be a separable real Hilbert space and let E be a separable real Banach space. We develop a general theory of stochastic convolution of ℒ(H,E)-valued functions with respect to a cylindrical Wiener process with Cameron-Martin space H. This theory is applied to obtain necessary and sufficient conditions for the existence of a weak solution of the stochastic abstract Cauchy problem (ACP) (t∈ [0,T]), almost surely, where A is the generator of a -semigroup of bounded linear operators on...
Let H be a separable real Hilbert space and let E be a real Banach space. In this paper we construct a stochastic integral for certain operator-valued functions Φ: (0,T) → ℒ(H,E) with respect to a cylindrical Wiener process . The construction of the integral is given by a series expansion in terms of the stochastic integrals for certain E-valued functions. As a substitute for the Itô isometry we show that the square expectation of the integral equals the radonifying norm of an operator which is...
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