## 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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### References:

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