On the stochastic Burgers’ equation in the real line. (English) Zbl 0939.60058

Gyöngy, István; Nualart, David

Ann. Probab. 27, No. 2, 782-802 (1999).

The authors consider a one-dimensional Burgers’ equation perturbed by a space-time white noise on the whole real line, instead of a bounded interval for the space variable. An existence and uniqueness theorem for the Cauchy problem is proved. The method is based on a suitable property of the semigroup corresponding to the Dirichlet problem for the heat equation in bounded space interval. This property is derived from a known regularization property of the semigroup via Sobolev’s embedding valid on bounded domains. The authors derive the property they need for the semigroup directly from well-known estimates on the heat kernel, which are valid not only in a bounded interval, but also in the whole real line. Using the corresponding estimates, the equivalence between the mild solution and the generalized solution defined via test functions in the integral form of the equations is established. Moreover, it is shown that the solution is continuous in the time and space variable \((t,x)\) if the initial value is continuous.

Reviewer: Anatoly Swishchuk (Kyïv)

MSC:

60H15	Stochastic partial differential equations (aspects of stochastic analysis)
60H20	Stochastic integral equations
60H40	White noise theory
35R60	PDEs with randomness, stochastic partial differential equations

Keywords:

stochastic Burgers’ equation; space-time white noise

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[14]	UNIVERSITY OF EDINBURGH GRAN VIA, 585 KING’S BUILDINGS 08007 BARCELONA EDINBURGH, EH9 3JZ SPAIN UNITED KINGDOM