We show that the homogeneous viscous Burgers equation $(partial_t-etaDelta) u(t,x)+(ucdot abla)u(t,x)=0, (t,x)in{mathbb{R}}_+times{mathbb{R}}^d$ $(dge 1, eta>0)$ has a globally defined smooth solution if the initial condition $u_0$ is a smooth function growing like $o(|x|)$ at infinity. The proof relies mostly on estimates of the random characteristic flow defined by a Feynman-Kac representation of the solution. Viscosity independent a priori bounds for the solution are derived from these. The regularity of the solution is then proved for fixed $eta>0$ using Schauder estimates. The result extends with few modifications to initial conditions growing abnormally large in regions with small relative volume, separated by well-behaved bulk regions, provided these are stable under the characteristic flow with high probability. We provide a large family of examples for which this loose criterion may be verified by hand.
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