Mathematics – Probability
Scientific paper
2007-08-27
Mathematics
Probability
v4: 53 pages & 1 table. Improvements in presentation and organization. Improved title. minor typo fixed. References updated. C
Scientific paper
We delve deeper into the compelling regularizing effect of the Brownian-time Brownian motion density, $\KBtxy$, on the space-time-white-noise-driven stochastic integral equation we call BTBM SIE, which we recently introduced. In sharp contrast to second order heat-based SPDEs--whose real-valued mild solutions are confined to $d=1$--we prove the existence of solutions to the BTBM SIE in $d=1,2,3$ with dimension-dependent and striking Holder regularity, under both less than Lipschitz and Lipschitz conditions. In space, we show a nearly local Lipschitz regularity for $d=1,2$--roughly, the SIE is spatially twice as regular as the Brownian sheet in these dimensions--and nearly local H\"older 1/2 regularity in d=3. In time, our solutions are locally H\"older continuous with exponent $\gamma\in(0,(4-d)/(8))$ for $1\le d\le3$. To investigate our SIE, we (a) introduce the Brownian-time random walk and we use it to formulate the spatial lattice version of the BTBM SIE; and (b) develop a delicate variant of Stroock-Varadhan martingale approach, the K-martingale approach, tailor-made for a wide variety of kernel SIEs including BTBM SIEs and the mild forms of many SPDEs of different orders on the lattice. Here, solutions types to our SIE are both direct and limits of their lattice version. The BTBM SIE is intimately connected to intriguing fourth order SPDEs in two ways. First, we show that it is connected to the diagonals of a new unconventional fourth order SPDE we call parametrized BTBM SPDE. Second, replacing $\KBtxy$ by the intimately connected kernel of our recently introduced imaginary-Brownian time-Brownian-angle process (IBTBAP), our SIE becomes the mild form of a linearized Kuramoto-Sivashinsky SPDE. Ideas developed here are adapted in separate papers to give a new approach, via our explicit IBTBAP representation, to many linear and nonlinear KS-type SPDEs in multi spatial dimensions.
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