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A basic identity for Kolmogorov operators in the space of continuous functions related to RDEs with multiplicative noise. (English) Zbl 1318.60068

The authors study the reaction-diffusion equation \[ \partial_t u(t,\xi)= \partial^2_{\xi}u(t,\xi) + f(\xi,u)+g(\xi,u)\partial_t w(t,\xi),\quad t\geq0,\;\xi\in[0,1], \] perturbed by a multiplicative space-time white noise \(\partial_t w(t,\xi)\) and appropriate boundary conditions \(u(t,0)=u(t,1)=0\), \(u(0,\xi)=x(\xi)\). The nonlinear term \(f\) is assumed to be of polynomial growth in the second variable. Analyzing the associated Kolmogorov operator and transition semigroup, they prove a modification of the identity known as identité du carré des champs in the classical additive noise case. As an application of this identity, they construct the Sobolev space \(W^{1,2}(E,\mu)\), where \(E=C_0([0,1])\) and \(\mu\) is an invariant measure for the system, and they prove the validity of the Poincaré inequality and of the spectral gap.

MSC:

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35R60 PDEs with randomness, stochastic partial differential equations
35R15 PDEs on infinite-dimensional (e.g., function) spaces (= PDEs in infinitely many variables)
35K57 Reaction-diffusion equations
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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