On the Stochastic Processes on $7$-Dimensional Spheres

We studied isometric stochastic flows of a Stratonovich stochastic differential equation on spheres, i.e. on the standard sphere and Gromoll-Meyer exotic sphere. The standard sphere $S^7_s$ can be constructed as the quotient manifold $\mathrm{Sp}(2, \mathbb{H})/S^3$ with the so-called ${\bullet}$-action of $S^3$, whereas the Gromoll-Meyer exotic sphere $\Sigma^7_{GM}$ as the quotient manifold $\mathrm{Sp}(2, \mathbb{H})/S^3$ with respect to the so-called ${\star}$-action of $S^3$. The Stratonovich stochastic differential equation which describes a continuous-time stochastic process on the standard sphere is constructed and studied. The corresponding continuous-time stochastic process and its properties on the Gromoll-Meyer exotic sphere can be obtained by constructing a homeomorphism $h: S^7_s\rightarrow \Sigma^7_{GM}$. The corresponding Fokker-Planck equation and entropy rate in the Stratonovich approach is also investigated.


Introduction
The microscopic description of the dynamics of a diffusion process on a connected compact differentiable manifold M, for instance, is typically represented by a so-called Stratonovich formulation of stochastic differential equations. Likewise, the dynamic descriptions of the other stochastic processes on differentiable manifolds are also represented in the formulation. The advantage of this formulation is that Itô's formula appears in the same form as the fundamental theorem of calculus; therefore, stochastic calculus in this formulation takes a more familiar form. The other advantage of Stratonovich formulation, which is more important and relevant to our investigation is that stochastic differential equations on manifolds in this formulation transform consistently under diffeomorphisms between manifolds.
In general, the laws of physics must be independent of the choice of a coordinate system. This statement means, for instance, that there must be a family of coordinate systems that are compatible with describing space-time. A set of compatible coordinate systems in space-time forms a differential structure. Space-time is an example of a topological space equipped with a differential structure. In a topological space, we may find more than one differential structures. Therefore, from a single topological space, we can construct more than one space-time with different differential structures. If the yielded space-times are not distinguishable in the sense that they are not diffeomorphic, they may lead to inequivalent formulations of the law of physics. The totality of inequivalent differential structures on a topological space is called the exotica on the topological space.
In mathematics, spheres are topological spaces which are interesting to investigate, and in many branches of physics, they serve for instance as models for configuration spaces of some mechanical systems. In physics, for example, the standard 7-dimensional sphere S 7 s is particularly interesting in related to supersymmetry breaking and to the work of Witten in which he used it to cancel the global gravitational anomalies in 1985. Two differential structures on 7-dimensional sphere S 7 are said to be equivalent if there is a diffeomorphism pulling back the differentiable maximal atlas from the second to the first. The connected compact topological space S 7 has more than one distinct differential structure that is not equivalent each other in this sense or more precisely there are connected compact 7-dimensional differentiable manifolds which are homeomorphic but not diffeomorphic to the standard seven-sphere S 7 s . The differential structure on the standard sphere S 7 s is called a standard differential structure, while the differential structures that are not equivalent to the standard one are called exotic differential structures. The topological space S 7 equipped with an exotic differential structure or a seven-dimensional differentiable manifold which is homeomorphic but not diffeomorphic to standard seven-dimensional sphere S 7 s is called exotic 7-sphere. Recently, Gromoll and Meyer have constructed a free action of S 3 on Sp(2, H) which preserves the bi-invariant metric. The quotient is an exotic 7-sphere. The first time where biquotients were considered in geometry, was in [4], where it was shown that an exotic 7-sphere admits non-negative curvature. The Gromoll-Meyer 7-sphere is the only exotic sphere which can be written as a biquotient. The concept of a biqoutient Sp(2, H)//Sp(1, H), is an exotic 7-sphere, which produced the first example of an exotic sphere with non-negative curvature. The non-negatively curved manifold Sp(2, H)//Sp(1, H) is homeomorphic, but not diffeomorphic, to S 7 s . The Gromoll-Meyer sphere Sp(2, H)//Sp(1, H) is obtained by letting Sp(1, H) act via diag-(q, q) on the left, and diag-(q, 1) on the right.
The properties of the Gromoll-Meyer exotic sphere Σ 7 GM related to the isometric stochastic flow are to be studied in this article. The properties of an isometric stochastic flow of the Stratonovich stochastic differential equation on the 7-dimensional spheres will be examined. The seven spheres which will be considered here are the standard sphere S 7 s and the Gromoll-Meyer exotic sphere Σ 7 GM . In this study, Σ 7 GM is understood to be the 7-dimensional sphere equipped with a Gromoll-Meyer (exotic) differential structure. The standard sphere S 7 s can be regarded as a quotient manifold Sp(2, H)/S 3 through •-action by S 3 and Σ 7 GM regarded as quotient manifold Sp(2, H)/S 3 but with respect to the ⋆-action of S 3 . A Stratonovich stochastic differential equation which describes the continuous stochastic process (i.e., the stochastic process which is parameterized with continuous-time) on the standard sphere, is constructed and examined. Regarding the continuous stochastic process formulated in Σ 7 GM , which is obtained by constructing a homeomorphism h : S 7 s → Σ 7 GM . The Fokker-Planck equation and entropy rate in the Stratonovich approach are also studied in both spheres.

Continuous stochastic process
Let (Ω, F , P) be a probability space, (E, E) a measurable space, and T a set. A mapping Z : is called a stochastic process with phase space (E, E) if the measurable maps (Z t ) t∈T : (Ω, F ) → (E, E), where (Z t ) t∈T (ω) = z t for every ω ∈ Ω, is a random field taking values in a measurable space (E, E). The sample paths of Z are the mappings Z(ω) : T → (E, E); t → (Z t ) t∈T (ω) obtained by fixing ω ∈ Ω. The sample paths of Z thus form a family of mappings from T into (E, E) indexed by ω ∈ Ω. A stochastic process Z = {(Z t ) t∈T (ω)} is said to be continuous, right-continuous, or left-continuous if for P-almost all ω, the sample path of ω is continuous, right-continuous, or left-continuous, respectively. A continuous-time stochastic process Z = {(Z t ) t∈T (ω)} is said to have continuous sample paths, if t → (Z t ) t∈T (ω) is continuous for all ω ∈ Ω [2].
The Brownian motion B = {(B t ) t∈T (ω)} with T = {0 ≤ t < ∞} is a continuous martingale, and a continuous local martingale can be (random) time changed to be a Brownian motion. Therefore, the study of Brownian motion has much to say about the sample path properties of much more general classes of processes, including continuous martingales and diffusion process.

Stochastic differential equation
In most applications, the phase space of a continuous-time stochastic process can be regarded as (R d , B), i.e., the d-dimensional Euclidean space equipped with the σ-algebra of Borel sets concerning the natural topology in R d . However, many applications are demanding another kind of measurable spaces (E, E) to play the role of phase space. In this study, we consider manifolds which are homeomorphic to S 7 s as the phase space of a continuous-time stochastic process.
The significant advantage of the Stratonovich stochastic integral is that it obeys the usual transformation rules of calculus. Stratonovich motivation was to achieve a rigorous treatment of the stochastic differential equation, which governs the diffusion processes of A. N. Kolmogorov. For this reason, Stratonovich formula is often used to formulate stochastic differential equations on manifolds such as a circle or sphere. Stochastic processes defined by Stratonovich integrals over a varying time interval does not, however, satisfy the powerful martingale properties of their Itô integral counterparts as we have already mentioned.
Let M be an (d − 1)-dimensional differentiable manifold embedded in R d . The usual continuous-time stochastic process on the manifold M is described by the following Stratonovich stochastic differential equation of the following form [6] where indicates that the integral involved being Stratonovich, and {U 0 , U 1 , · · · , U (d−1) } is a C ∞ -vector field on the manifold M. Equation (1) has a solution flow g s,t : Ω × M → M, consisting of random smooth diffeomorphisms of M, continuous in t.
The last term of equation (1) can be regarded as a noise or a random disturbance adjoined to the ordinary differential equation dz t = U 0 (z t )dt. Here the continuous sample paths t → B i t (ω) are not functions of bounded variations with respect to t a.s, so that dB i t (ω) cannot be defined as the Stieltjes integral. Nevertheless the last integrals are well defined for almost all samples if z t is adapted, i.e., for any t, z t is independent of the future Brownian motion B s (ω) − B t (ω), s ≥ t. The Stratonovich stochastic differential equation (1) can also be written more concisely as Let z (n) = (z 1 , z 2 , · · · , z n ) be n-point in M or an element of M. Set g s,t (ω, z (n) ) = (g s,t (ω, z 1 ), g s,t (ω, z 2 ), · · · , g s,t (ω, z n ), t ≥ s). Then for each fixed s ≥ 0 and z (n) , g s,t (ω, z (n) ), t ∈ [s, T ] is a continuoustime stochastic process with values in M starting at z (n) at time s. It is called an n-point motion of the flow g s,t (ω). Stochastic flows on manifolds are defined similarly to those on Euclidean space.
Let M be a smooth (d − 1)-dimensional manifold and G be a Lie group of diffeomorphisms M → M. Suppose φ t is a diffusion on G with φ 0 = I, I the identity of G. φ t will be called a G-valued stochastic flow on M. For any x ∈ M, φ t (x) is a stochastic process on M with φ 0 (x) = x and will be called the one-point motion of φ t starting from x. There has been an extensive effort over the past few years to study the stochastic flow of smooth random fields, g, from a general Lie group to Riemannian manifold M [7].
3 Isometric stochastic flows and Gromoll-Meyer exotic sphere

Isometric stochastic flows
If M is endowed with a Riemannian metric g, an action l : G × M → M of a group G on (M, g) is said to be isometric (or by isometries) if g : M → M is an isometry of (M, g) for all g ∈ G. In this case, the metric g is said to be G-invariant, and G can be identified with a subgroup of Iso(M, g). A G-valued stochastic flow will be called an isometric stochastic flow, if and only if M is a Riemannian manifold and G is a Lie group of isometries. In order for the stochastic flow generated by equation (1) to be isometric (the equation (1) is contained in the isometry group G of M) and associated to Brownian motion on M, the vector fields U i in (1) must be Killing vector fields and the sum of , on M such that they form the Laplace-Beltrami operator, then there is an isometric stochastic flow whose one point motion is a Brownian motion on M [7]. Therefore, a mapping g s,t (ω, ·) : x ∈ M → g s,t (ω, x) = x t ∈ M is an isometric stochastic flows whose one point motion is a Brownian motion on M, i.e., a diffusion process generated by equation (1) [6]. The map g s,t (ω) will be referred to as the isometric stochastic flows associated to (1). Furthermore, the isometric stochastic flows act naturally on Killing vector fields.
We consider the standard 7-sphere, isometrically embedded as the unit sphere in a 8-dimensional Euclidean vector space (V, g): be a special orthogonal group acting on S 7 s ⊂ R 8 . The isometric action of SO(8, R) on R 8 is transitive on the unit sphere. The solution of a Stratonovich stochastic differential equation on SO(8, R) is an SO(8, R)-valued isometric stochastic flow g s,t (ω) on S 7 s . In fact, U 0 = 0 and U 1 , U 2 , · · · , U 7 is vector fields on S 7 s of unit speed rotations determined by 2-dimensional coordinate planes in R 8 . The vector fields on S 7 s are the Killing vector field that forms Laplace-Beltrami operator of S 7 s . Therefore, there is an isometric stochastic flow g s,t (ω) on S 7 s whose one point motion is a Brownian motion. Hence, the motion of S 7 s can be described by an isometric stochastic flow g s,t (ω) on S 7 s . The motion of a fixed point on S 7 s is a Brownian motion x t on S 7 s , will be called the one point motion x t of g s,t (ω). The set of maps g s,t (ω) is called an isometric stochastic flow of homeomorphisms if there exists a null set N of Ω so that for any ω ∈ N c , the set of continuous maps {g s,t (ω) : s, t ∈ [0, T ]} defines an isometric stochastic flow of homeomorphisms, i.e., it satisfies the following properties [6] ( with no exceptions, where • denotes the composition of maps. If we use the notation g s,t (ω, z) instead of g s,t (ω)(z), for z ∈ S 7 s , then (1) becomes: Loosely speaking, each random field g s,t is a random transformation of S 7 s , and (2) is a kind of semigroup property, often called the flow property. A Gaussian random field is a random field where any finite collection of elements g s,t (z 1 ), · · · , g s,t (z 8 ) has a multidimensional Gaussian distribution.
(3) the map g s,t (ω, ·) : S 7 s → S 7 s is an onto homeomorphism for all s, t ∈ [0, T ]. This g s,t is called an isometric stochastic flow of homeomorphisms generated by the Killing vector field Further, if g s,t (ω) satisfies (4) g s,t (ω, z) is k-times differentiable with respect to z for all s, t and the derivatives are continuous in (s, t, z), Then it is called an isometric stochastic flow of C k diffeomorphisms.
Let g s,t (ω) −1 be the inverse map of g s,t (ω). Then the conditions (1) and (2) imply g t,s (ω) = g s,t (ω) −1 , s ≤ t. This fact and the condition (3) show that g s,t (ω) −1 (z) is also continuous in (s, t, z). The condition (4) implies that g s,t (ω) −1 (z) is k-times continuously differentiable with respect to z. Hence g s,t (ω, ·) : S 7 s → S 7 s is actually a C k -diffeomorphism for all s, t if (4) is satisfied. We can regard g s,t (ω) −1 (z) as a random field with parameter (s, t, z), and denotes it as g −1 s,t (ω, z). Thus g −1 s,t (ω, z) = g t,s (ω, z) holds for all s, t, z a.s. An isometric stochastic flow of homeomorphisms can be considered as a continuous random field with values in SO(8, R) satisfying the flow properties (1) and (2). We will call it a continuous isometric stochastic flow with values in SO(8, R) [6].  H) is subduced onto Σ 7 GM to provide Σ 7 GM automatically with a Riemannian metric of non-negative sectional curvature (K ≥ 0). Every bi-quotient of the compact Lie transformation group of quaternionic unitary matrices has a Riemannian metric of non-negative sectional curvature and thereby Σ 7 GM is the first exotic sphere with non-negative sectional curvature.
In 2002 Totaro [9] and independently Kapovitch and Ziller [5] showed that Σ 7 GM is the only exotic sphere that can be expressed as a bi-quotient of a compact Lie transformation group. Note that Σ 7 GM can be regarded as the basic example of a bi-quotient in Riemannian geometry and, simultaneously as the core example of an exotic sphere [9]. In fact, Σ 7 GM is diffeomorphic to the Milnor exotic sphere Σ 7 2,−1 . There are two free isometric actions of S 3 on Sp(2, H) (which are isometric for many metrics and connections) that plays a central role in the rest of the paper. Both of these actions foliate Sp(2, H) by S 3 -orbits in two different ways. The first action is the standard action given as follows : if q ∈ S 3 and Q ∈ Sp(2, H) [4], where The action therefore leads to a principal fibration S 3 → Sp(2, H) → S 7 s called •-fibration. The orbit space of the standard •-action can be naturally identified with S 7 s ⊂ H 2 by restricting a matrix in Sp(2, H) to its first column [8]. Let π • be the projection map of the above principal bundle with the base space being the S 7 s . The projection is therefore given by The second one is the Gromoll-Meyer exotic action given by [4] The second action leads to a principal fibration S 3 → Sp(2, H) → Σ 7 GM called ⋆-fibration. The base manifold of this fibration is the well-known Σ 7 GM .

Main Results and Discussion
4.1 Stochastic processes on S 7 s As we have mentioned before that the class a b c d can be identified with (b, d) ∈ H×H satisfying bb+dd = 1, where b = b 0 +b 1 i+b 2 j+b 3 k and d = d 0 +d 1 i+d 2 j+d 3 k. Now consider the unit sphere S 7 s as the submanifold of R 8 , i.e., a S 7 is given as a set of points in R 8 whose coordinates (z 1 , z 2 , · · · , z 8 ) Let (b, d) be any point on S 7 s . The point (b, d) can be identified for instance with a point z = (z 1 , z 2 · · · , z 8 ) ∈ R 8 , as follows Since bb + dd = 1, (10) Therefore, the point z = (z 1 , z 2 , z 3 , · · · , z 8 ) ∈ R 8 which is identified with (b, d) is in S 7 s ⊂ R 8 .

Orthonormal vector field on S 7
s An interesting fact of S 7 s is that we can find a globally defined frame on it, i.e. globally nonvanishing vector fields whose values at every point form a basis for the tangent space. From short observation, it is clear that the tangent space at a point z ∈ S 7 s is the 7-dimensional subspace of R 8 consisting of all vectors which are perpendicular to z. In particular every vector field on S 7 s can be represented by a continuous function v : s . In this way, the linear structure of vector fields become even more apparent. The idea is to assign to each point z ∈ S 7 s a vector v(z) tangent to z in a smooth way. However, by assumption v(z) is nonvanishing, so we can normalize such that v(z) . Therefore, there exist seven nonvanishing linearly independent vector fields on S 7 s . For every point z = (z 1 , z 2 , · · · , z 8 ) ∈ S 7 s , the following vector fields on S 7 s are nonvanishing and linearly independent [3]: (11) It can be shown that the frame {U 1 , U 2 , · · · , U 7 } is left invariant and orthonormal at each point z ∈ S 7 s , with respect to the Euclidean metric in R 8 .

Stratonovich stochastic differential equations on S 7 s
The Stratonovich stochastic differential equation on S 7 s that we will construct in this part is more general than equation (1), in the sense that we will have as integrator not only a Brownian motion but also an arbitrary continuous semimartingale W . Continuous semimartingales are a useful (and widely used) tool for two non-independent reasons: many processes that can be defined explicitly are continuous semimartingales (for instance, diffusion processes); many operations applied to continuous semimartingales yield continuous semimartingales. The space of continuous semimartingales is an appropriate frame for stochastic calculus [2].
All the vector fields containing in the frame we have discussed earlier, i.e., {U 1 , U 2 , U 3 , · · · , U 7 }, are Killing vector fields on S 7 s , since the vector fields of the form are Killing and the fact that the Lie derivative of the metric g satisfies L V +W g = L V g + L W g for arbitrary vector fields V and W . For those vector fields in the frame, we obtain the following systems of Stratonovich stochastic differential equations dz µ t = U µ (z t ) • dW t , (µ = 1, 2, · · · , 7) (15) or Since U 1 , U 2 , · · · , and U 7 are Killing vector fields, the flows generated by equation (15) are isometric. The isometric stochastic flowsz µ t will be called frame isometric stochastic flows. Due to equation (16), the processz µ t is a semimartingale for all µ = 1, · · · , 7.
An arbitrary vector field A on S 7 s can be written as a linear combination of the frame {U 1 , · · · , U 7 }, namely A = A µ U µ (Einstein convention of summation). Therefore, the isometric stochastic flow z t generated by A is given by (17) From equation (15), the last expression can be written as The above vector field A is a Killing vector field if L A µ Uµ g = 0, i.e., Since U µ (µ = 1, 2, · · · , 7) are Killing vector fields, then A is also Killing vector field if for all i, j = 1, 2, · · · , 8. Therefore, the stochastic flow generated by equation (18) is isometric if the vector fields A satisfies equation (20).

4.1.3
Fokker-Planck equations on S 7 s associated to frame isometric stochastic flow and their entropy rate Equation (15) can be written in matrix notation as follows: where U 1 µ , U 2 µ , · · · , U 8 µ are the components of U µ . The Stratonovich integral of stochastic differential equation (21) is given by Note that as long as the diffusion function δ i k U i µ (z t ) is only dependent on the random field then Itô and Stratonovich interpretations of the stochastic differential equation are the same. The modified drift function (Itô-Stratonovich correction) in equation (22) is given by (e.g., [1]) In fact, the Itô counterpart of the Stratonovich stochastic differential equation is a useful artifice which, for example, allows access to the Fokker-Planck equation. Similarly, we must use the corresponding Itô stochastic differential equation to determine the appropriate coefficients of the Fokker-Planck equation for a diffusion process arising from the solution of a Stratonovich equation.
Therefore, if a physical system, on the one hand, is defined by the Stratonovich stochastic differential equations (21) then the same process can also be described by the Itô equation: with the solution is given bȳ Itô-Stratonovich stochastic differential equation and Fokker-Planck equations can be formulated to describe the dynamical system in any coordinate of a Riemannian manifold in a way that is very similar to the case of R d . Therefore, the derivation of the Fokker-Planck equation governing the time evolution of conditional probability density function on S 7 s proceeds in an analogous way to the derivation of such equation in R 8 . In general, the volume of S 7 s will be an 7-dimensional measure.
where U 1 µ A µ , U 2 µ A µ , · · · , U 8 µ A µ (without summation in index µ) are the components of a matrix U A, (U A) iµ = U i µ A µ . The Stratonovich integral of stochastic differential equation (35) is given by