Kindred Spirits in Mathematics

Atsushi Atsuji (Professor, Department of Mathematics)

I believe the term "martingale" has become more familiar recently, partly due to its use in fields like financial engineering. A martingale is a type of stochastic process that possesses the following property.

The left side represents the conditional expectation given the information \( F_s \) about \( M_t \) up to time \( S \). In other words, if we predict the future at time \( T \) and calculate the expected value based on what is known up to the present time \( S \), the result will be the current value. A typical example is the profit gained from a fair game. Mathematically, typical examples include simple random walks and Brownian motion, but martingales represent a much broader class of stochastic processes that includes these. Doob discovered that this stochastic process is similar to harmonic functions, which are well known in function theory. If Δ is the Laplacian, a function that satisfies Δ\(u=0 \) is called a harmonic function. Looking only at these two definitions, they do not seem similar at all, but if we apply the optional stopping theorem to the martingale,

In the case of the real plane, applying the mean value theorem (for harmonic functions) to a harmonic function gives,

This means that taking the average at time \( T \) (on the circumference of a circle with radius \( r \)) yields the value at time 0 (the center of the circle). You might think that mathematicians must be a truly carefree lot to be delighted by such a superficial resemblance. The important point is that the relationship is not just superficial; they are closely related in their mathematical structure, almost the same thing. In fact, it turns out that a harmonic function can be considered almost a martingale. This requires an important intermediary: Brownian motion. If we consider \( u(X_t) \), which is the result of substituting Brownian motion \( X_t \) on a plane into a harmonic function \( u \), this becomes a martingale. In other words, the class of stochastic processes called martingales can be seen as containing the class of functions called harmonic functions. This means that to study harmonic functions, one can study martingales. Since Doob, martingale theory has been actively researched, and much has been learned about harmonic functions using arguments from probability theory, including martingale theory. I should also add that while Brownian motion served as an intermediary, the theory of diffusion processes, including Brownian motion, plays a major role in such research. This has made it possible to study harmonic functions on singular spaces such as manifolds and fractal sets.

Now, let me introduce another pair of look-alikes from a completely different field.

On the complex plane \( C \), a function whose derivative with respect to the complex conjugate of z = x + \( \sqrt{-1} \) y is 0 is called a holomorphic function. A function expressed as the ratio of two holomorphic functions with no common zeros is called a meromorphic function. Nevanlinna theory is a theory that investigates the value distribution of meromorphic functions. Vojta proposed (in 1987) that this Nevanlinna theory is similar to Roth's theorem, which is known in the Diophantine approximation theory of number theory. Let's look at a part of it.

"R. Nevanlinna's Second Main Theorem (1925):

For any non-constant meromorphic function \( f \) and distinct points \( a_1,…,a_q\)ϵC∪{∞},

holds for all 0< \( r \) outside a set of finite length."

Here, \( m\)(\(r, a_k\)) is a function that measures the "proximity" of the image of \( f \) to the point \( a_k \), and \( T_f \)(\(r\)) is a function that measures the "size" of the image of \( f \),

and satisfies \( T_f \)(\( r \)) ↑∞ as \( r \)↑∞.

"Roth's Theorem (1955): If \( α \) is an algebraic number, then for any ϵ>0,

holds for any coprime integers \( m,n \) (\( n\)>0), with a finite number of exceptions."

An algebraic number is a number that is a solution to a polynomial equation with rational coefficients. The left side represents the "proximity" when approximating \( α \) with the rational number \( m/n \), and the inequality shows that this proximity is limited by a certain "size" of the rational number on the right side. It certainly looks like a similar functional relation, and a "2" appears on the right side. Unlike the case of martingales and harmonic functions, there is no direct connection conceivable here, but many commonalities are known. Incidentally, L. Ahlfors discovered that the "2" on the right side of Nevanlinna's theorem corresponds to the Euler characteristic of the sphere (≅C∪{∞}). The above can be stated more generally; the similarity between the equations of Diophantine approximation, a focus of number theory, and the fundamental inequalities of function theory has stimulated many mathematicians and promoted research. Since Vojta, there have been many attempts to tackle number theory problems through the study of holomorphic maps, and this discovery has had a major impact on complex analysis, complex geometry, and number theory.

I myself am focusing on the relationship between Nevanlinna theory and holomorphic martingales, which are known as the probabilistic analogues of holomorphic and meromorphic functions. Using this, it is possible to show that an analogue of Nevanlinna's Second Main Theorem holds for meromorphic functions on a wide class of complex manifolds. Furthermore, in relation to Vojta's discovery, my speculations continue to grow: what about random holomorphic functions? What is the relationship with the optimality of Nevanlinna theory? Is there a familiar "intermediary" between Diophantine approximation and Nevanlinna theory? Speculation alone achieves nothing, of course. In any case, I believe that finding these "look-alikes" from completely different fields is a driving force that moves mathematics forward.