Abstract: |
A stochastic partial differential equation (s.p.d.e.) is a mathematical tool used to model a physical phenomenon perturbed by a random noise. In the classical theory, the noise is supposed to be white in both time and space, and the solution is understood in the sense of distributions. Recently, Robert Dalang (Electr. J. Probab. 1999) developed the mathematical theory of s.p.d.e.’s driven by colored noises. Such equations are mathematically more challenging, but they may arise in the context of more general physical situations.
In this talk, we will consider the stochastic heat equation in (0; T) £ Rd, driven by a Gaussian noise which is white in time, and colored in space. The covariance structure of the noise is given by a kernel function f, which is the Fourier transform of a tempered distribution in Rd. Particular cases are the Bessel kernel of order ® > 0 and the Riesz kernel of order ® 2 (0; d). In both cases, it is known that if ® > d ¡ 2, then the equation possesses a solution u = fu(t; x); (t; x) 2 (0; T) £ Rdg which is a multi-parameter Gaussian process. In this talk, we will discuss the germ Markov property of the process u, using a characterization of its Reproducing Kernel Hilbert Space and a classical criterion for Markovianity in the Gaussian case, due to Pitt (1971), or Kunsch (1979).
In the case of the Bessel kernel of order ®, we are able to show that the process u is germ Markov if ® = 2k; k 2 N+; similarly, for the Riesz kernel, we have a positive answer only in the case ® = 4k; k 2 N+. To prove that the germ Markov property of the process u holds (or does not hold) for Bessel and Riesz kernels of arbitrary index ® (or for more general kernel functions) remains an open problem.
This talk is based on joint work with Doyoon Kim.
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