## The heat equation and reflected Brownian motion in time-dependent domains.(English)Zbl 1046.60060

The reflecting Brownian motion on a time-space $$C^3$$-domain is constructed by solving the corresponding Skorokhod stochastic differential equation. Existence, uniqueness and strong Markov property are proved for the solution. Moreover, the strong Feller property of the heat semigroup as well as a Gaussian type upper bound of the heat kernel are provided. As applications, probability representations are presented for heat equations with mixed time-space boundary conditions. Finally, for the one-dimensional case, various properties concerning the heat kernels or the distributions of the reflecting Brownian motion are obtained for time-space domains with merely continuous boundary.

### MSC:

 60H30 Applications of stochastic analysis (to PDEs, etc.) 35K05 Heat equation 35K20 Initial-boundary value problems for second-order parabolic equations
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### References:

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