MATHEMATICS

E. B. Dynkin

DIFFUSION OF TENSORS

(Presented by Academician A. N. Kolmogorov, 12 VI 1967)

1. Let an affine connection \(\Gamma_{jk}^{i}\) be given on a smooth \(l\)-dimensional manifold \(E\) of class \(C^3\), and let \(X\) be a diffusion process on \(E\) with generating differential operator

\[ \mathfrak{D}=a^{ij}\nabla_i\nabla_j+b^i\nabla_i . \tag{1} \]

(\(\nabla_i\) is the covariant differentiation corresponding to the connection \(\Gamma_{jk}^{i}\).) The totality \(R_m^n(x)\) of all tensors of valence \((m,n)\) attached at the point \(x\in E\) forms a linear space. The union of \(R_m^n(x)\) over all \(x\in E\) will be denoted by \(R_m^n\). It will be shown that the process \(X\) induces a diffusion process \(U\) in the space \(R_m^n\). Denote by \(L_m^n\) the space of linear continuous functions on \(R_m^n\).*

It is shown that the semigroup of linear operators \(T_t\) corresponding to the process \(U\) leaves the space \(L_m^n\) invariant. The infinitesimal operator \(A\) of the semigroup \(\mathcal{T}_t\) induced in \(L\) by the semigroup \(T_t\) is computed. The result can be described as follows. Let \(u\in R_n^m\) and \(w\in R_m^n\). Put
\[ (u,w)=u^{j_1\ldots j_n}_{i_1\ldots i_m}w^{i_1\ldots i_m}_{j_1\ldots j_n}. \]
Every function \(f\) from \(L_m^n\) can be described by the formula

\[ f(u)=(u,w(x))\quad \text{for } u\in R_m^n(x), \tag{2} \]

where \(w(x)\) is some tensor field of valence \((m,n)\). This defines a natural identification of the space \(L_m^n\) with the space \(\mathscr{L}_m^n\) of continuous tensor fields on \(E\) of valence \((n,m)\). The infinitesimal operator \(A\) acts on such fields according to the formula

\[ Aw(x)=a^{ij}(x)\nabla_i\nabla_j w(x)+b^i(x)\nabla_i w(x), \tag{3} \]

i.e. it formally has the same form as the operator \(\mathfrak{D}\) (see (1)).

2. Put

\[ \Gamma_\alpha u^{j_1\ldots j_n}_{i_1\ldots i_m} = \sum_\mu \Gamma^{\beta}_{\alpha j_\mu} u^{j_1\ldots j_n}_{i_1\ldots i_{\mu+1}\beta i_{\mu+1}\ldots i_m} - \sum_\nu \Gamma^{i_\nu}_{\alpha\beta} u^{j_1\ldots j_{\nu-1}\beta j_{\nu+1}\ldots j_n}_{i_1\ldots i_n}. \tag{4} \]

It is easy to verify that

\[ (\Gamma_\alpha u,w)=-(u,\Gamma_\alpha w)\quad (u\in R_m^n(x),\ w\in R_n^m(x)). \tag{5} \]

Let \(y(s)\) \((s_0\le s\le s_1)\) be a smooth curve. A family of tensors \(u(s)\in R_m^n(y_s)\) is called parallel if

\[ \dot u(s)=(\Gamma_\alpha u(s))\dot y^\alpha(s). \tag{6} \]

Integrating the differential equation (6) with the initial condition \(u(s_0)=h\), we define the parallel transport of a vector \(h\in R_m^n(y(s_0))\) along the curve \(y(s)\). Omitting the argument \(s\), we shall write the equation—

* By a tensor of valence \((m,n)\) we mean a tensor of the type
\[ u^{j_1\ldots j_n}_{i_1\ldots i_m}, \]
i.e. a tensor \(m\) times co- and \(n\) times contravariant.

** A function defined on \(R_m^n\) is called linear if it is linear on each space \(R_m^n(x)\) \((x\in E)\).

…equation (6) in abbreviated form

\[ \dot u = (\Gamma_{\alpha}u)\dot y^{\alpha}. \tag{7} \]

1. The diffusion process \(X\) corresponding to the generating operator \(\mathfrak{D}\) can be constructed as the solution of the system of stochastic differential equations of Ito

\[ dx^{\alpha}(t)=\rho^{\alpha}[x(t)]\,dt+\sigma_{\beta}^{\alpha}[x(t)]\,d\xi^{\beta}(t), \tag{8} \]

where \(\xi^{1}(t),\ldots,\xi^{l}(t)\) are \(l\) independent Wiener processes. It is known (see, for example, \(({}^{1})\), Sec. 11.12) that the generating differential operator of the process \(x(t)\) is given by the formula

\[ \mathfrak{D}f=\rho^{\alpha}\partial_{\alpha}f+\tfrac12\tilde a^{\alpha\beta}\partial_{\alpha}\partial_{\beta}f, \tag{9} \]

where

\[ \tilde a^{\alpha\beta}=\sum_{\gamma=1}^{l}\sigma_{\gamma}^{\alpha}\sigma_{\gamma}^{\beta}. \tag{10} \]

The covariant derivative \(\nabla_{\alpha}\) is related to the partial derivative \(\partial_{\alpha}\) by the relation

\[ \nabla_{\alpha}=\partial_{\alpha}-\Gamma_{\alpha}. \tag{11} \]

Therefore, for

\[ \rho^{\alpha}=b^{\alpha}-a^{\beta\gamma}\Gamma_{\beta\gamma}^{\alpha},\qquad \tilde a^{\alpha\beta}=2a^{\alpha\beta} \tag{12} \]

the operator defined by formula (9) coincides with the operator (1).

1. We now define stochastic parallel transport of tensors along the trajectory \(x(t)\) of the diffusion process \(X\). Consider on this trajectory two nearby points \(x(t)\) and \(x(t+\Delta t)\) and join them by a smooth curve \(y(s)\). According to Sec. 2, a family of tensors \(u(s)\), parallel along the curve \(y(s)\), satisfies equation (7). We shall agree to write \(Q_1\approx Q_2\) if \(Q_1-Q_2=o(\Delta s^2)\). We have

\[ \Delta u=u(s+\Delta s)-u(s)\approx \dot u\Delta s+\tfrac12\ddot u\Delta s^2. \tag{13} \]

On the other hand,

\[ \Delta x^i=x^i(t+\Delta t)-x^i(t)=y^i(s+\Delta s)-y^i(s)\approx \dot y^i\Delta s+\tfrac12\ddot y^i\Delta s^2. \]

Hence

\[ \dot y^{\alpha}\dot y^{\beta}\Delta s^2\approx \Delta x^{\alpha}\Delta x^{\beta},\qquad \tfrac12\ddot y^i\Delta s^2\approx \Delta x^i-\dot y^i\Delta s. \]

Denote by \(\Pi_{\alpha\beta}\) the operator obtained if, on the right-hand side of (4), \(\Gamma_{\alpha j}^{\,i}\) is replaced by \(\partial_{\beta}\Gamma_{\alpha j}^{\,i}\); in other words, put

\[ \Pi_{\alpha\beta}u=(\partial_{\beta}\Gamma_{\alpha})u. \]

Differentiating equation (7) with respect to \(s\) and substituting the values of \(\dot u\) and \(\ddot u\) into (13), we obtain

\[ \Delta u\approx (\Pi_{\alpha\beta}u+\Gamma_{\alpha}\Gamma_{\beta}u)\Delta x^{\alpha}\Delta x^{\beta}+\Gamma_{\alpha}u\Delta x^{\alpha}. \tag{14} \]

The right-hand side does not depend on the choice of the smooth curve joining \(x(t)\) and \(x(t+\Delta t)\). This choice can affect only the value of \(\Delta s\) and, consequently, the meaning of the relation \(\approx\). We restrict the choice of curves \(y(s)\) only by the requirement that \(\sum_i(\Delta x^i)^2/\Delta s^2\) be bounded above and below by positive constants as \(\Delta t\to 0\). It is easy to verify that, if this requirement is satisfied in some local coordinate system at the point \(x(t)\), then it is also satisfied in any coordinate system smoothly related to the original one. We may therefore take as \(y(s)\) curves defined by linear equations in some admissible coordinate system (one could also have taken a segment of a geodesic). Now consider a subdivision \(0=t_0<t_1<\ldots<t_n=t\) of the interval \([0,t]\) and write equation (14) for each interval \([t_i,t_{i+1}]\). Passing to the limit as \(\max |t_i-t_{i-1}|\to 0\)

and, taking into account that \(x(t)\) is given by equation (8), we arrive at the following stochastic differential equation defining stochastic parallel transport:

\[ du=(\Pi_{\alpha\beta}u+\Gamma_\alpha\Gamma_\beta u)\,dx^\alpha dx^\beta+(\Gamma_\alpha u)\,dx^\alpha . \tag{15} \]

Here \(dx^\alpha\) is expressed by formula (8), while \(dx^\alpha dx^\beta\), by definition, is equal to \(\tilde a^{\alpha\beta}dt\). Substituting these values in (15), we arrive at the equation

\[ du=\{\Pi_{\alpha\beta}u+\Gamma_\alpha\Gamma_\beta u\}\tilde a^{\alpha\beta} +\varrho^\alpha\Gamma_\alpha u]dt+(\Gamma_\alpha u)\sigma^\alpha_\gamma d\xi^\gamma . \tag{16} \]

The system of stochastic differential equations (8) and (16) defines a diffusion process \(U\) in \(R_m^n\). The rank of the diffusion matrix of this process does not exceed the rank of \(a^{\alpha\beta}\). Therefore the process \(U\) is strongly degenerate, even if the process \(X\) is nondegenerate.

5. Let \(h\in L_m^n(x_0)\). Denote by \(x_h(t), u_h(t)\) the solution of the system of equations (8), (16) under the initial condition \(x_h(0)=x_0,\ u_h(0)=h\). Since \(x_h(t)\) is the point of application of the tensor \(u_h(t)\), \(x_h(t)\) is uniquely recovered from \(u_h(t)\).

The semigroup \(T_t\) of the process \(U\) is defined by the formula

\[ T_t f(h)=\mathbf M f[u_h(t)] . \tag{17} \]

It is easy to see that \(u_h(t)\) is linear in \(h\). Therefore the space \(L_m^n\) is invariant with respect to \(T_t\). Identifying, by means of formula (2), the space \(L_m^n\) with the space \(\mathscr L_n^m\), we note that the semigroup \(T_t\) induces in \(\mathscr L_n^m\) a semigroup \(\mathscr T_t\), acting according to the formula

\[ (h,\mathscr T_t w(x))=\mathbf M\,(u(t),w[x(t)]) . \tag{18} \]

According to Itô’s calculus of stochastic differentials (see \(({}^2)\) or \(({}^1)\), Ch. 7),

\[ d(u,w(x))=(u,dw(x))+(du,w(x))+(du,dw(x)), \tag{19} \]

where \(du\) is defined by formula (15) or (16), and

\[ dw^i(x)=\partial_\alpha w^i(x)\,dx^\alpha+\tfrac12\partial_\alpha\partial_\beta w^i(x)\,dx^\alpha dx^\beta . \tag{20} \]

Expressing \(\partial_\alpha w\) and \(\partial_\alpha\partial_\beta w\) with the aid of formula (11) and relying on (5), we obtain

\[ (u,\partial_\alpha w)=(u,\nabla_\alpha w)-(\Gamma_\alpha u,w), \]

\[ \begin{aligned} (u,\partial_\alpha\partial_\beta w) &=(u,\nabla_\alpha\nabla_\beta w)-(\Gamma_\beta\Gamma_\alpha u,w)-(\Pi_{\beta\alpha}u,w) \\ &\quad-(\Gamma_\beta u,\partial_\alpha w)-(\Gamma_\alpha u,\partial_\beta w) +\Gamma^\gamma_{\alpha\beta}(u,\nabla_\gamma w). \end{aligned} \tag{21} \]

From (19), (20), and (21),

\[ d(u,w(x))=(u,\nabla_\alpha w)dx^\alpha -\tfrac12 (u,\nabla_\alpha\nabla_\beta w+\xi\Gamma^\gamma_{\alpha\beta}\nabla_\gamma w)dx^\alpha dx^\beta . \]

Substituting here the value of \(dx^\alpha\) from (8) and taking (12) into account, we have

\[ d(u,w(x))=(u,\mathfrak D w(x))dt+(u,\nabla_\alpha w(x))\sigma^\alpha_\varrho d\xi^\varrho . \]

Integrating from \(0\) to \(t\), taking the mathematical expectation and using (18), we have

\[ (h,\mathscr T_t w(x))-(h,w(x)) =\int_0^t \mathbf M\,(u(s),\mathfrak D w[x(s)])\,ds . \]

Hence it follows easily that, for tensor fields of class \(C^2\), \(Aw=\mathfrak D w\), i.e. formula (3) is valid.

The general theory of semigroups (see, for example, (1), Chap. 1) makes it possible to conclude that, for \(w(x) \in C^2\), the tensor field \(w_t = \mathscr{T}_t w\) satisfies the differential equation

\[ \partial w_t / \partial t = \mathfrak{D} w_t \]

with the initial condition \(w_0 = w\).

1. What has been presented is a further development of the investigations begun by K. Ito in (3). Ito considered a more special case, in which: a) in formula (1) \(b^i = 0\), and \(a^{ij}\) admits an inverse matrix \(a_{ij}\); b) the connection \(\Gamma_{jk}^{i}\) is the Riemannian connection (without torsion) corresponding to the metric tensor \(a_{ij}\); c) the tensors \(u\) have valence \((0,n)\). Ito’s definition of stochastic parallel displacement also differs somewhat.

Moscow State University
named after M. V. Lomonosov

Received
12 VI 1967

REFERENCES

\(^{1}\) E. B. Dynkin, Markov Processes, Moscow, 1963.
\(^{2}\) K. Ito, Nagoya Math. J., 3, 55 (1951); Collection of translations Matematika, 3, 5, 131 (1959).
\(^{3}\) K. Ito, Proc. Intern. Congress Math., 15–22 VIII 1962, Uppsala, 1963, p. 536.