We prove a weighted L p -estimate for the stochastic convolution associated with the stochastic heat equation with zero Dirichlet boundary condition on a planar angular domain Dκ0⊂R2 with angle κ∈ (0 , 2 π). Furthermore, we use this estimate to establish existence and uniqueness of a solution to the corresponding equation in suitable weighted L p -Sobolev spaces. In order to capture the singular behaviour of the solution and its derivatives at the vertex, we use powers of the distance to the vertex as weight functions. The admissible range of weight parameters depends explicitly on the angle κ.
The first author has been supported by the Deutsche Forschungsgemeinschaft (DFG, Grant DA 360/20-1) and partially by the Marsden Fund Council from Government funding, administered by the Royal Society of New Zealand. The collaboration between the authors has been significantly facilitated by a travel grant of the German Academic Exchange Program (DAAD) and the National Research Foundation of Korea (NRF) within the DAAD-NRF Bilateral Scientist Exchange Program, supporting a two-month visit of the first author to the second and the third author. The second author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (NRF-2014R1A1A2055538). The third author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (NRF-2013R1A1A2060996).
