Uniform in time modulus of continuity of Brownian motion

Let $B=(B_t{)}_{t\ge 0}$ be a standard Brownian motion. The main objective is to find a uniform (in time) control of the modulus of continuity of $B$ in the spirit of what appears in (Kurtz, 1978). More precisely, it involves the control of the exponential moments of the random variable ${\sup }_{0\le s\le t}\mathrm{\mid }{B}_t-B_s\mathrm{\mid }\mathrm{/}w(t,\mathrm{\mid }t-s\mathrm{\mid })$ for a suitable function $w$. A stability inequality for diffusion processes is then derived and applied to two simple frameworks.