Brownian-time Brownian motion SIEs on $\mathbb{R}_{+}$ × $\mathbb{R}^d$: Ultra regular direct and lattice-limits solutions and fourth order SPDEs links
Brownian-time Brownian motion SIEs on $\mathbb{R}_{+}$ × $\mathbb{R}^d$: Ultra regular direct and lattice-limits solutions and fourth order SPDEs links
We delve deeper into the compelling regularizing effect of the Brownian-time Brownian motion density,$ \mathbb{K}^{{BTBM}^d}_{t;x,y}$ , on the space-time-white-noise-driven stochastic integral equation we call BTBM SIE:\begin{equation} U(t,x)=\int_{{\mathbb R}^{d}}{{\mathbb K}}^{\text{BTBM}^d}_{t;x,y} u_0(y) dy+ \int_{{\mathbb R}^{d}}\int_0^t{{\mathbb K}}^{\text{BTBM}^d}_{t-s;x,y} a(U(s,y))\mathscr W(ds\times dy), (0.1)\end{equation} which we recently introduced in [3].In sharp contrast to traditional second order …