covariant stochastic calculus in the sense of itô

Volume 109A, number 5 PHYSICS LETTERS 27 May 1985

COVARIANT STOCHASTIC CALCULUS IN TH E SENSE OF ITO

Robert GRAHAM

Fachbereich Physik, Universitiit Essen, Essen, West Germany

Received 15 March 1985; accepted for publication 23 March 1985

The Langevin equation in the sense of It6 on a manifold is put into a manifestly covariant form, which reduces to the original form in harmonic coordinates and directly corresponds to the covariant Fokker-Planck equation and a covariant form of the functional integral in preproint discretization.

Stochastic differential equations on a manifold with local coordinates qV are usually formulated either in the sense of It6 [1 ] ,

dq v = KV(q)dt + g~(q), dW / , (1)

or in the sense of Stratonovich [2],

dq ~ = fV(q)dt + g~.(q)o dW i . (2)

Here dW i is the increment of an n-dimensional Wiener process with

(dW i) = 0, (dWidW ]) = 5i/dt. (3)

The It6 product g~/(q) • dW i and the Stratonovich product g~(q) o dW i are, respectively, defined by [1,2]

~ ( q ) . dWi(t) = tim g~(q(t)) [Wi(t + At) - Wi(t)], (4) At.-.0 +

g~(q)o dWi(t) = lira ~[gr(q(t + At)) + g~(q(t))] [Wi(t + At) - Wi(t)] . (S) Ate0

Furthermore, in eqs. (1), (2)

KV(q) =fV(q) + ~ [ag'[(q)/aqU] ~(q)8* (6)

and

QVU(q ) = g~ (q ) g~k (q ) 8~ (7)

are, respectively, the drift coefficients and the diffusion matrix appearing in the equivalent Fokker-Planek equa. tion for the probability density P,

ae_ _L at aqV ~ aqVi}q u Q='U(q)p. (8)

For simplicity it is assumed in most of the following that the inverse of QVU exists. From a practical point of view the advantage of It6's calculus is that q~(r) and dWi(t) are deeorreiated for r < t and therefore KV(q) and QVU(q) can be immediately read off from It6's equation. The disadvantage is the unconventional transformation rule for dq v in It6's calculus. For q,V = q,V(q) we have

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1 &i'v = (aq,v/~lU). &iu + .~ (a2q,~/aqUaqa) QUa dt . (9)

As a conscquence,g~(q) but not &iv and K v in eq. (1) transform as contravariant vectors. In Stratonovich's product, on the other hand, dWi(t) and qV(t) are eorrehted, but &iv transforms as a contravariant vector,

dq 'v = (Oq'V/OqU) o &iu , (10)

and so does) w [3] which is, of course, an important advantage in all problems involving riemannian manifolds. However, an important disadvantage of Stratonovich's calculus is that eqs. (2), (6) are not invariant under q-de- pendent rotations of dW i and g~ involving Stratonovich's product:

dW; -~ d ~ i = ~Z~(q)o dW/, v - , _ " , gi(q) -+ g i (q) - ~li(q) g) (q) , ( l l )

with

while eqs. (1), (7), (8) are invariant under such rotations if It6's product is used in

dW i ~ d[~ i = a}(q)" dWJ, gr(q)-~ f r ( q ) = ~ ( q ) g;(q) . (13)

Therefore, dqV,JW(q) and g~(q) transform as contravariant vectors in Stratonovich's calculus, but only at the price that Stratonovich's drift vector i f (q ) contains a spurious part, which has nothing to do with the Fokker-Planck equation. In fact, two Stratonovich drift vectorsfV(q) andl'V(q) rehted by

~,V(q) = fV(q) _ ~ [o~2k (q)/~qU ] ~2~(q) 6il g~(q) g~(q) (14)

with arbitrary ~.(q) satisfying eq. (12) correspond to the same Fokker-Planck equation. The arbitrary degrees of freedom of the drift vector and its possible uses have been analyzed in detail in ref. [4]. Clearly, it would be de- sirable to have a manifestly eovariant stochastic differential equation but to avoid introducing any spurious quantities. It appears of great interest, therefore, to cast It6's calculus into a manifestly covariant form, which we now wish to do.

Eq. (9) shows that dq v does not transform as a contravariant vector in It6's calculus. However, it is possible to extract from dq ~ the contravariant part Dq v by the definition

I v Kh DqV=dqV+~F xQ dt , (15)

where F v is the affine connection associated with the metric tensor QVU Explicitly, with Q = det QUV we have gh

Dq v = & l v - } V ~ a - ~ (QVX/V~) d t . (16) Oq

Then It6's stochastic differential equation (1) acquires the manifestly covariant form

DCl v = hV(q) dt + g~(q) " dW i . (17)

Here

hV(q) = KV(q) + ,} 1-~ x QgX (18)

is the contravariant drift vector [3] appearing in the covariant form

aS/~t = (hvS - ~ QVU S~); v (19)

of the Fokker-Planek equation (8). S = x/'-QP is the invariant probability density, and covariant derivatives in eq. (19) are denoted by semicolons.

The contravariant differential Dq v coincides with dq v in harmonic coordinates [3] which axe def'med by the condition b(QVX]~)[bqX = O. On the other hand different harmonic coordinates are connected by transforma- tions q'V = q,V(q) for which (O2q'V/bqUOqX) = O. Therefore, in It6's calculus &iv already transforms as a contra-

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variant vector within the restricted class of harmonic coordinate systems. The fully contravariant extension Dq v of dq v is uniquely defined as that contravariant vector, which coincides with dq v in harmonic coordinates.

The functional integral representation of the conditional probability densityP(q Iqo, t) in It6's calculus takes the simplest possible form (see ref. [5] )

t

P(qlqo,') = fD[q] /D[p/21rlS(q(t)-q)exp(fdr{}Q"U(q).p..pu-pv. [4q"/d¢-K"(q)]}), (20)

where the measure of integration

D[ql O[p/2~rl = lira 1-I dq"(t/)~.(t/)/27r (21) N~** j = l -

is invariant within It6's calculus. The integral over p(r) is taken along the imaginary axis and the integral over q(r) is taken only over paths q(r) with q(0) = q0" The discretization associated with It6's calculus is the "prepoint discretization" (see ref. [5] ) where t

f ,dr{} Q"(q) "p." Pt~ -P~" [dq"/d¢ - KV(q)] ) 0

N-1

= lim ~ At{} QVU(q(ri))p,(ri+l)pu(ri+l) _ pv(ri+l) [(q,(ri+l) _ q,(ri))/A t _ K"(q(r/))] ) . (22) N--*** i=0

NAt=t

The functional integral (20) is not manifestly covariant. However, using the contravariant differential (15) it can now be cast into the manifestly covariant form

S(qlqo,t) = f D[ql f D[pl2~l "drQ(q)6(q(t)-q)exp d'd.½ QVU(q)" Pv " P , - P v " [~"l~-h"(q)l} , (23)

where

S(q Iq o, t) = X/'-~)P(q Iqo, t) . (24)

In (23) Pv, Dq v, hu and QUU transform as vectors and tensors, respectively. Again prepoint discretization is associated with (23). For the contravariant DqV(t)/dt tiffs entails the discrete representation

• ( ) DqU(r/) -* qV(¢/+l) - qU(r/) - } x / r ~ ) • ~-~ [QVX(q)/Qx/Q~)] q =q(~/)

Eq. (23) gives a manifestly covariant functional integral representation of the invariant conditional probability density in It6's calculus. By integration over p(¢) (assuming that Q.., the inverse of Quu, exists) it is reduced to the equally covariant form

S(qlqo,t) = f Dtz[ql Qx/-O-~)8(q(t)-q)exp(- f d r } Q , u ( q )" [DqU/Dr-hU(q)] [Dq"/Dr-hU(q)]) , (26)

where N

D/~[q] = lira I I dnq(ri)[(2nAt) n Q(ri_l) ] -1/2 (27) N--*** i=1

N ,X t=t is a measure of integration in path space which is invariant in It6's calculus, as can be seen from its covariant representation

D/.t[q] ; D [ q , f D[p/27r] exp(} )drQVU(q) .pu .pu ) . (28, 0

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Finally, we remark that it is not necessary to use the tensor Q~V(q) as a metric tensor. A different choice is more useful in particular ff the inverse of QUU does not exist and the system instead is endowed with another natural metric tensor gUU(q ). Then F~, in eq. (15) and eq. (18) is the affme connection expressed by gV~(q), S is defined by S = (Det gu~)l/2p, and eq. (19) is replaced by

aS/Or : - [h~S - ½ (QpuS);~] ;v- (29)

The functional integrals (23), (26) retain their form except for the change of Q-+ det guy. The integration measure (27) remains unchanged, but it exists only if QVU(q) is non-singular. Otherwise only eq. (23) can be used. Also in

v Kh DqlZ this more general case there exists a special class of coordinates in which FKX Q = 0 and coincides with dqU. In summary, eqs. (17), (19), (23) combine, the advantages of a manifestly eovariant representation with the

usual advantages of It0's calculus - i.e. maximal simplicity of the relation between the stochastic differential equation, the Fokker-Planck equation and the functional integral - and the avoidance of any spurious quantities.

References

[1] K. It6, Proc. Imp. Acad. 20 (1944) 519; K. It6 and S. Watanabe, in: Stochastic differential equations, ed. K. It6 (Wiley, New York, 1978).

[2] R.L. Stratonovich, Conditional Markov processes and their application to the theory of optimal control (Elsevier, New York, 1968).

[3] R. Graham, Z. Phys. B26 (1977) 397. [4] D. Ryter and U. Deker, J. Math. Phys. 21 (1980) 2662;

U. Deker and D. Ryter, J. Math. Phys. 21 (1980) 2666. [5 ] F. Langouche, D. Roekaerts and E. Tirapegui, Functional integration and semi-classical expansions (Reidel, Dordrecht, 1982).

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covariant stochastic calculus in the sense of itô

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