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Itô’s- and Tanaka’s-type formulae for the stochastic heat equation: The linear case. (English) Zbl 1084.60039
The authors study the stochastic heat equation with additive noise in dimension one, that is, \(dX_t = \Delta X_t dt + dW_t,\) \(t \in (0,T], X_0=0\), where \(W\) denotes a cylindrical Brownian motion. They understand the solution of such equation in a mild sense and can be solved explicitly in the form of a stochastic convolution. Using the representation of the solution given by the convolution of \(W\) by the operator-valued kernel \(e^{(t-s)\Delta}\), the authors obtain Itô’s and Tanaka’s type formulae associated to \(X\). In order to obtain these results, they adapt the methodology used to study some properties of fractional Brownian motion to their infinite-dimensional case.

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60H05 Stochastic integrals
60H07 Stochastic calculus of variations and the Malliavin calculus
60G15 Gaussian processes
Full Text: DOI
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