  In this paper we prove global bounds on the spatial gradient of viscosity
solutions to second order linear and nonlinear parabolic equations in $(0,T)
\times \R^N$. Our assumptions include the case that the coefficients be both
unbounded and with very mild local regularity (possibly weaker than the Dini
continuity), the estimates only depending on the parabolicity constant and on
the modulus of continuity of coefficients (but not on their $L^{\infty}$-norm).
Our proof provides the analytic counterpart to the probabilistic proof used in
Priola and Wang (J. Funct. Anal. 2006) to get this type of gradient estimates
in the linear case. We actually extend such estimates to the case of possibly
unbounded data and solutions as well as to the case of nonlinear operators
including Bellman-Isaacs equations. We investigate both the classical
regularizing effect (at time $t>0$) and the possible conservation of Lipschitz
regularity from $t=0$, and similarly we prove global H\"older estimates under
weaker assumptions on the coefficients. The estimates we prove for unbounded
data and solutions seem to be new even in the classical case of linear
equations with bounded and H\"older continuous coefficients. Applications to
Liouville type theorems are also given in the paper. Finally, we compare in an
appendix the analytic and the probabilistic approach discussing the analogy
between the doubling variables method of viscosity solutions and the
probabilistic coupling method.
