Abstract
We consider time-inhomogeneous, second-order linear parabolic partial differential equations of the non-divergence type, and assume the ellipticity and the continuity on the coefficient of the second-order derivatives and the boundedness on all coefficients. Under the assumptions, we show the Hölder continuity of the solution in the spatial component. Furthermore, additionally assuming the Dini continuity of the coefficient of the second-order derivative, we have the better continuity of the solution. In the proof, we use a probabilistic method, in particular the coupling method. As a corollary, under an additional assumption we obtain the Hölder and Lipschitz continuity of the fundamental solution in the spatial component.
Original language | English |
---|---|
Pages (from-to) | 1063-1088 |
Number of pages | 26 |
Journal | Journal of Evolution Equations |
Volume | 17 |
Issue number | 3 |
DOIs | |
Publication status | Published - Sep 1 2017 |
Keywords
- Coupling method
- Diffusion
- Fundamental solution
- Hölder continuous
- Lipschitz continuous
- Parabolic partial differential equation
- Stochastic differential equation
ASJC Scopus subject areas
- Mathematics (miscellaneous)