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This paper provides a probabilistic proof of the comparison result for viscosity solutions of path-dependent semilinear PDEs. We consider the notion of viscosity solutions introduced in [I. Ekren, et al., Ann. Probab., 42 (2014), pp. 204-236], which considers as test functions all those smooth processes which are tangent in mean. When restricted to the Markovian case, this definition induces a larger set of test functions and reduces to the notion of stochastic viscosity solutions analyzed in [E. Bayraktar and M. Sirbu, Proc. Amer. Math. Soc., 140 (2012), pp. 3645-3654; SIAM J. Control Optim., 51 (2013), pp. 4274-4294]. Our main result takes advantage of this enlargement of the test functions and provides an easier proof of comparison. This is most remarkable in the context of the linear path-dependent heat equation. As a key ingredient for our methodology, we introduce a notion of punctual differentiation, similar to the corresponding concept in the standard viscosity solutions [L. A. Caffarelli and X. Cabre, Amer. Math. Soc. Colloq. Publ., 43, AMS, Providence, RI, 1995], and we prove that semimartingales are almost everywhere punctually differentiable. This smoothness result can be viewed as the counterpart of the Aleksandroff smoothness result for convex functions. A similar comparison result was established earlier in [I. Ekren et al., Ann. Probab., 42 (2014), pp. 204-236]. The result of this paper is more general and, more importantly, the arguments that we develop do not rely on any representation of the solution. [ABSTRACT FROM AUTHOR]

Consider a rooted tree T with leaf-set [n] and with all nonleaf vertices having out-degree 2, at least. A rooted tree T with leaf-set S⊂[n] is induced by S in T if T is the lowest common ancestor subtree for S, with all its degree-2 vertices suppressed. A "maximum agreement subtree" (MAST) for a pair of two trees T′ and T′′ is a tree T with a largest leaf-set S⊂[n] such that T is induced by S both in T′ and T′′. Bryant, McKenzie, and Steel [BioConsensus, AMS, Providence, RI, 2003, pp. 55-65] and Bernstein et al. [SIAM J. Discrete Math., 29 (2015), pp. 2065-2074] proved, among other results, that for T′ and T′′ being two independent copies of a random binary (uniform or Yule-Harding distributed) tree T, the likely magnitude order of MAST(T′,T′′) is O(n1/2). We prove this bound for a wide class of random rooted trees: T is a terminal tree of a branching, Galton-Watson, process with an ordered-offspring distribution of mean 1, conditioned on "total number of leaves is n." [ABSTRACT FROM AUTHOR]