BTP SIEs on $\Rp\times\Rd$: Ultra regular BTRW SIEs limits solutions, the K-martingale approach, and fourth order SPDEs links

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