We introduce a class of interesting stochastic processes based on Brownian-time processes. These are obtained by taking Markov processes and replacing the time parameter with the modulus of Brownian motion. They generalize the iterated Brownian motion (IBM) of Burdzy and the Markov snake of Le Gall, and they introduce new interesting examples. After defining Brownian-time processes, we relate them to fourth order parabolic partial differential equations (PDE's). We then study their exit problem as they exit nice domains in d , and connect it to elliptic PDE's. We show that these processes have the peculiar property that they solve fourth order parabolic PDE's, but their exit distribution-at least in the standard Brownian time process case-solves the usual second order Dirichlet problem. We recover fourth order PDE's in the elliptic setting by encoding the iterative nature of the Brownian-time process, through its exit time, in a standard Brownian motion. We also show that it is possible to assign a formal generator to these non-Markovian processes by giving such a generator in the half-derivative sense.

The prevalent wisdom on developing membrane active antimicrobials (MAAs) is to seek a delicate, yet unquantified, cationic-hydrophobic balance. Inspired by phages that use nanostructured protein devices to invade bacteria efficiently and selectively, we study here the antibiotic role of nanostructures by designing spherical and rod-like polymer molecular brushes (PMBs) that mimic the two basic structural motifs of bacteriophages. Three model PMBs with different well-defined geometries consisting of multiple, identical copies of densely packed poly(4-vinyl-N-methylpyridine iodide) branches are synthesized by controlled/"living" polymerization, reminiscent of the viral structural motifs comprised of multiple copies of protein subunits. We show that, while the individual linear-chain polymer branch that makes up the PMBs is hydrophilic and a weak antimicrobial, amphiphilicity is not a required antibiotic trait once nanostructures come into play. The nanostructured PMBs induce an unusual topological transition of bacterial but not mammalian membranes to form pores. The sizes and shapes of the nanostructures further help define the antibiotic activity and selectivity of the PMBs against different families of bacteria. This study highlights the importance of nanostructures in the design of MAAs with high activity, low toxicity, and target specificity.

We report here a green chemistry method to conjugate hydrophobic payloads (Lilial as a prototype) highly efficiently (35.8 wt %) with (1f4)-2-amino-2-deoxy-β-D-glucan (i.e., chitosan) via Schiff base bond formation in an ionic liquid, which renders chitosan easily dissolvable in common organic solvents and amenable to further functional modifications. As an example, thermoresponsive poly(N-isopropylacrylamide) was grafted to the chitosan-Lilial conjugate. The graft copolymer self-assembled in water at neutral pH into core-shell nanocarriers with a favorable size distribution (d ∼ 142 ( 60 nm) for intravenous administration. Under conditions of enhanced temperature and acidity (T = 37°C, pH = 4.5) mimicking endosomal or lysosomal uptake, the nanocarriers fell apart and formed reversed micelles with greatly reduced sizes (d ∼ 8 ( 3 nm) favoring clearance by renal filtration, and 70% Lilial molecules were liberated within 30 h through hydrolytic cleavage of the exposed Schiff base conjugation. This smart stimuli-responsive drug release profile reveals a viable approach in the development of chitosan-based nanocarriers for intravenous administration of hydrophobic pharmaceuticals.

We show that the optimal exercise boundary for the American put option with non-dividend-paying asset is convex. With this convexity result, we then give a simple rigorous argument providing an accurate asymptotic behavior for the exercise boundary near expiry.KEY WORDS: American put option on a zero dividend asset, convexity of the early exercise boundary, free boundary problem, obstacle problem, near-expiry estimates.