Stochastic areas

Posted on August 24, 2023 by Fabrice Baudoin

Nizar Demni, Jing Wang and I finished the preliminary draft of a monograph to be later published by the European Mathematical Society (EMS Tract Series).

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The study of area type functionals associated with Brownian motions has interested me for a long time. By its simplicity, the number of its far reaching applications, and its connections to many areas of mathematics, the Paul Levy‘s stochastic area formula is undoubtedly among the most important and beautiful formulas in stochastic calculus.

Let Z(t)=B_1(t)+iB_2(t), t\ge0 , be a Brownian motion in the complex plane such that Z(0)=0. The algebraic area swept out by the path of Z up to time t is given by half of the value

S(t)=\int_{Z[0,t]} (xdy-ydx)=\int_0^t \left(B_1(s) dB_2(s) -B_2(s)dB_1(s)\right),

where the stochastic integral is an Ito integral, or equivalently a Stratonovich integral since the quadratic covariation between B_1 and B_2 is 0. The Levy’s area formula


\mathbb{E}\left( e^{i\lambda S(t)} | Z(t)=z\right)=\frac{\lambda t}{\sinh \lambda t} e^{-\frac{|z|^2}{2t}(\lambda t \coth \lambda t -1) }


was originally proved by P. Levy using a series expansion of Z. The formula nowadays admits many different proofs. A particularly elegant probabilistic approach is due to Marc Yor. The first observation is that, due to the invariance by rotations of the law of the path of Z, one has for every \lambda \in \mathbb R,
\mathbb{E}\left( e^{i\lambda S(t)} | Z(t)=z\right)=\mathbb{E}\left( \left. e^{-\frac{\lambda^2}{2} \int_0^t |Z(s)|^2 ds} \right| |Z(t)|=|z| \right).
One considers then the new probability measure
\mathbb{P}{/ \mathcal{F}_t}^\lambda = \exp \left( \frac{\lambda}{2}(|Z(t)|^2 -2t) -\frac{\lambda^2}{2} \int_0^t |Z(s)|^2 ds \right)\mathbb{P}{/ \mathcal{F}_t}

under which, thanks to Girsanov theorem, (Z(t))_{t \ge 0} is a Gaussian process (an Ornstein-Uhlenbeck process). The area formula then easily follows from standard computations on Gaussian measures.

Somewhat surprisingly the Levy area formula and the stochastic area process (S(t))_{t \ge 0} appear in many different contexts, for instance, among many other references:

  1. S. Watanabe points out in this paper the connection with the differential of the exponential map in Lie groups;
  2. The formula also appears in the work by J.M. Bismut where probability methods are used to prove index theorems.
  3. The Mellin transform of S(t) is closely related to analytic number theory and in particular to the Riemann zeta function, see the beautiful survey paper by P. Biane, J. Pitman and M. Yor.
  4. The stochastic area process is a central character in the Terry Lyons’ rough paths theory.
  5. The formula also appears as an important tool in Malliavin calculus, see in particular the paper by S. Watanabe.
  6. The stochastic area process is also intimately connected to sub-Riemannian geometry. More precisely, in his paper, B. Gaveau actually observed that the 3-dimensional process (B_1(t),B_2(t), S(t))_{t\ge0} is a horizontal Brownian motion on Heisenberg group. As a consequence, the Levy area formula yields an expression for the heat kernel of the sub-Laplacian on the Heisenberg group.

The stochastic area process and the Levy area formula can be generalized in many different directions. For instance there exist analogues for Gaussian processes as in the paper by N. Ikeda, S. Kusuoka, and S. Manabe.

In the monograph we present generalizations of both the area process and the Levy area formula for Brownian motions on manifolds. As we show, natural generalizations of the stochastic area for a Brownian motion (X(t))_{t \ge 0} on a manifold \mathbb M are functionals that write S(t)=\int_{X[0,t]} \alpha

where \alpha is a one-form with some geometric significance taking values in a Lie algebra (or one of its quotients). To ensure the existence of an explicit Levy area type formula we will need that \mathbb M is a Riemannian homogeneous space and that \alpha is a form on \mathbb M coming from the connection form of a homogeneous bundle over \mathbb M. For instance, in the simplest case where \alpha is \mathbb R-valued, the area one-form will arise from a Kahler structure. By definition, a Kahler form on a complex manifold \mathbb M is a closed 2-form \omega that induces the metric on \mathbb M in the sense that \omega (X,JY) is the Riemannian metric on \mathbb M where J is the almost complex structure. It is a classical result in complex analysis that such 2-form can (at least locally) be written as \omega =i\partial \bar{\partial} \Phi where \partial and \bar{\partial} are the Dolbeault operators and \Phi is a smooth function. The (at least locally defined) real one-form
\alpha=\frac{1}{2i}(\partial -\bar{\partial}) \Phi
is then a natural area one-form on \mathbb M which satisfies d \alpha=(\partial +\bar{\partial}) \alpha=\omega. When \mathbb M is compact the homogeneous bundle over \mathbb M we mentioned before is a \mathbb S^1-bundle referred to as the Boothby-Wang fibration. It is worth noting that the pull-back to that bundle of the form \alpha then yields a winding one-form: Integrating this form against a path describes the \mathbb S^1 fiber component of that path.

We also consider more general one-forms \alpha like \mathfrak{su}(2)-valued one-forms and associated bundles, however we stress that our goal in the monograph is not to develop a general and abstract theory of stochastic area type functionals associated with homogeneous bundles. We rather focus on specific relevant examples for which very concrete calculations can be done. Covering in great details specific examples will give us the opportunity to explore several topics of independent interest related to the study of stochastic area functionals. We will in particular focus our attention on connections with the theory of Riemannian submersions and associated horizontal Brownian motions, the theory of complex and quaternionic projective and hyperbolic spaces, the theory of hypoelliptic heat kernels and the theory of random matrices.

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