If one accepts that probability theory is part of statistics, then
certainly statistics has contributed mightily to parts of mathematics.
Indeed, the theory of classical potential theory is in a real sense
equivalent to the theory of Brownian motion. Also, other Markov
processes have their own associated potential theories. These matters
are covered in Joseph Doob's Springer-Verlag book Classical Potential
Theory and Its Probabilistic Counterpart. The interplay between potential
theory and probability theory has led to new discoveries in both fields
and enriched both ends of the connection. Along the same lines, there
are many connections between Feynman's path integral approach to quantum
physics and Wiener function space integrals via Brownian motion. An
account of this can be found in Barry Simon's Academic Press book
Functional Integration and Quantum Physics. There is a probabilistic
proof of the Atiyah-Singer Index Theorem. There are connections between
complex analysis and Brownian motion and the stochastic approach to Hardy
H^p spaces was a minor industry for a while. Refer to Richard Durrett's
book Brownian Motion and Martingales in Analysis (Wadsworth publisher).
Differential geometry has been attacked successfully via stochastic
methods; refer to the work of Paul Malliavan and others. Probability
theory is an integral part of many parts of analysis and partial
differential equations. This is a major story of the past 40 years or so.
Richard Griego, Retired Professor of Mathematics,
University of New Mexico