03.09.29 · modern-geometry / spin-geometry

The probabilistic heat kernel and Bismut's formula

shipped3 tiersLean: none

Anchor (Master): Bismut 1984 *The Atiyah-Singer theorems: a probabilistic approach I, II* (J. Funct. Anal. 57); Berline-Getzler-Vergne *Heat Kernels and Dirac Operators* (Grundlehren 298) Ch. 11; Hsu *Stochastic Analysis on Manifolds* (GSM 38); Varadhan 1967 *On the behavior of the fundamental solution of the heat equation with variable coefficients* (Comm. Pure Appl. Math. 20); Ikeda-Watanabe *Stochastic Differential Equations and Diffusion Processes* (North-Holland)

Intuition Beginner

Heat spreads. Drop a hot spot at one point of a curved surface and watch the temperature even out. There is a second way to picture the same process that turns out to be far more powerful: instead of tracking temperature, release a tiny particle that jitters around at random, a drunken walker with no memory. After a short time the walker is somewhere nearby, with a bell-shaped spread of likely positions. The probability of finding it at a given spot is exactly the temperature that the heat process would have deposited there. Heat flow and random walking are two faces of one coin.

This is the bridge between analysis and probability. The heat kernel, the function $p_{t} (x, y)$ that says how much heat reaches $y$ in time $t$ from a unit hot spot at $x$ , is the same thing as the probability density for the random walker to travel from $x$ to $y$ in time $t$ . To compute heat, you can instead average over all the wandering paths the particle might take.

A remarkable fact follows. For very short times, the most likely paths are the straight ones, the shortest routes across the surface. So watching where the walker lands after a brief moment secretly measures geodesic distance. The random walker is a surveyor.

Now give the walker something to carry: a little arrow, or a spinning gyroscope, that it must keep pointing as straight as it can while the ground curves underneath. The arrow gets twisted around by the curvature of the surface. This is the picture Bismut built in 1984 for spinors: a random path that drags a spinor along, accumulating curvature, and the average over all such paths reproduces the heat flow of the Dirac operator, the square-root of the Laplacian that governs spin.

Visual Beginner

Picture a sphere with a starting dot at the north and a target dot lower down. From the start, draw a fuzzy cloud of jagged random paths, all wandering but on average drifting toward the target. Overlay one smooth geodesic arc, the great-circle route, in bold. The short-time story is that the cloud of paths presses in tightly around that bold arc: the random walker, to reach a far point quickly, has almost no choice but to follow the shortest road.

Along the bold path, draw a small frame of two perpendicular arrows at several points, each rotated a little from the last. That rotation is parallel transport: the arrows stay as straight as the curved surface permits, yet curvature still turns them. The total turning recorded by the frame, when the carried object is a spinor rather than a plain arrow, is the curvature term in Bismut's formula.

The second panel shows the bell curve. Above the target, plot the height $p_{t} (x, y)$ for shrinking $t$ : the bump sharpens and its logarithm, scaled by $2 t$ , settles onto the squared distance. The shape of heat at short times is the shape of distance.

Worked example Beginner

Take the simplest curved-free case first, the flat line, so the geometry does not get in the way. A random walker starts at $x = 0$ and takes independent random steps. After time $t$ its position is spread out in a bell curve centred at $0$ with width growing like the square root of $t$ . The probability density of landing at $y$ is

p_{t} (0, y) = \frac{1}{2 π t} e^{- y^{2} / (2 t)} .

This is the heat kernel of the flat line, and it is also the law of the random walker. Two readings, one formula.

Now read off the short-time fact by hand. Take the logarithm and multiply by $2 t$ :

2 t lo g p_{t} (0, y) = 2 t lo g \frac{1}{2 π t} - y^{2} .

As $t$ shrinks to zero the first term fades next to the second, and what survives is $- y^{2}$ . On the flat line the distance from $0$ to $y$ is just $∣ y ∣$ , so $- y^{2} = - d (0, y)^{2}$ . The scaled logarithm of the heat kernel converges to minus the squared distance. That is Varadhan's law in its baby form, and it holds on any curved surface once $∣ y ∣$ is read as geodesic distance.

To average a quantity $f$ over the walker, you weight $f (y)$ by this density and add up. In symbols the heat-flowed value of $f$ at the start point is the expected value of $f$ at the walker's random endpoint, $E_{0} [f (X_{t})]$ . Computing heat becomes computing an average.

Check your understanding Beginner

Formal definition Intermediate+

Let $(M, g)$ be a complete Riemannian manifold with Laplace-Beltrami operator $Δ$ (the geometer's non-positive Laplacian, so $- Δ \geq 0$ ). Brownian motion on $M$ is the diffusion process $X_{t}$ whose generator is $\frac{1}{2} Δ$ ; equivalently, $X_{t}$ is the Eells-Elworthy-Malliavin solution of a stochastic differential equation built by horizontally lifting a Euclidean Brownian motion to the orthonormal frame bundle $O (M)$ and projecting down (this construction is developed in 03.02.45). Its transition density with respect to the Riemannian volume is exactly the heat kernel $p_{t} (x, y)$ , the fundamental solution of $\partial_{t} u = \frac{1}{2} Δ u$ .

The basic identity is the probabilistic representation of the heat semigroup,

(e^{t Δ/2} f) (x) = E_{x} [f (X_{t})] = \int_{M} f (y) p_{t} (x, y) d vol (y),

valid for bounded measurable $f$ , with $E_{x}$ the expectation over Brownian paths started at $x$ .

Varadhan's short-time asymptotic sharpens the small- $t$ behaviour:

t \to 0 lim 2 t lo g p_{t} (x, y) = - d (x, y)^{2},

where $d$ is the Riemannian distance. The heat kernel sees the metric.

For a Schrodinger operator $\frac{1}{2} Δ - V$ with potential $V$ , the Feynman-Kac formula inserts an exponential weight along the path:

(e^{t (Δ/2 - V)} f) (x) = E_{x} [f (X_{t}) exp (- \int_{0}^{t} V (X_{s}) d s)] .

For a connection Laplacian on a vector bundle $E$ with curvature, the scalar weight is replaced by an $End (E)$ -valued multiplicative functional, carried by stochastic parallel transport; this is the Feynman-Kac-Ito formula, the subject of the key theorem below.

Key theorem with proof Intermediate+

Theorem (Feynman-Kac-Ito for the Bochner Laplacian). Let $E \to M$ be a Hermitian vector bundle with metric connection $\nabla^{E}$ , and let $Δ^{E} = - (\nabla^{E})^{*} \nabla^{E}$ be the connection (Bochner) Laplacian. Let $F \in End (E)$ be a self-adjoint bundle endomorphism. Then the heat semigroup of $\frac{1}{2} Δ^{E} + F$ admits the representation

(e^{t (Δ^{E} /2 + F)} s) (x) = E_{x} [M_{t} / /_{t}^{- 1} s (X_{t})],

where $/ /_{t} : E_{x} \to E_{X_{t}}$ is stochastic parallel transport along the Brownian path, and $M_{t} \in End (E_{x})$ is the multiplicative functional solving the pathwise covariant ordinary differential equation $\frac{d M _{t}}{d t} = M_{t} (/ /_{t}^{- 1} F / /_{t})$ , $M_{0} = id$ .

Proof. Fix a smooth section $s$ and set $u (t, x) = (e^{t (Δ^{E} /2 + F)} s) (x)$ , the solution of the bundle heat equation $\partial_{t} u = \frac{1}{2} Δ^{E} u + F u$ with $u (0, \cdot) = s$ . Pull $u$ back along the Brownian path into the fixed fibre $E_{x}$ by forming $N_{t} = M_{t} / /_{t}^{- 1} u (t - \cdot, X_{\cdot})$ evaluated at time $t$ ; concretely consider $Y_{r} = M_{r} / /_{r}^{- 1} u (t - r, X_{r})$ for $0 \leq r \leq t$ . Apply the Ito formula for the manifold diffusion. The second-order Ito term produces precisely $\frac{1}{2} Δ^{E}$ acting through the parallel transport (this is the defining feature of the horizontal lift: the generator of $/ /_{r}^{- 1} u (X_{r})$ is the connection Laplacian), the drift term contributes $- \partial_{t} u + \frac{1}{2} Δ^{E} u$ , and the multiplicative functional contributes $+ F u$ through its defining equation. The deterministic parts cancel by the heat equation, leaving $d Y_{r} = (martingale increment)$ . So $Y_{r}$ is a local martingale; under the standard completeness and growth hypotheses it is a genuine martingale. Equating expectations at $r = 0$ and $r = t$ gives $u (t, x) = Y_{0} = E_{x} [Y_{t}] = E_{x} [M_{t} / /_{t}^{- 1} s (X_{t})]$ , which is the claim. $□$

Bridge. This representation builds toward Bismut's probabilistic index theorem and is the foundational reason the curvature of a Clifford module enters the heat kernel as a parallel-transported, path-ordered exponential rather than as an explicit Green's function. This is exactly the bundle-valued upgrade of the scalar Feynman-Kac formula 02.15.04, and it generalises the Lichnerowicz decomposition $D^{2} = \nabla^{*} \nabla + \frac{1}{4} Scal + F^{E / S}$ of 03.09.08 into stochastic form: the connection Laplacian becomes Brownian motion with parallel transport, while the zeroth-order curvature term $\frac{1}{4} Scal + F^{E / S}$ becomes the multiplicative functional $M_{t}$ . The central insight is that putting these together turns the small-time limit of the supertrace into a Brownian-bridge integral that localises on constant loops, which appears again in the Master tier as Bismut's alternative to the Getzler rescaling of 03.09.20 — the same index density, reached by conditioning paths instead of rescaling coordinates.

Exercises Intermediate+

Exercise 1 (medium, proof). Show that the multiplicative functional for a constant endomorphism $F$ commuting with parallel transport reduces to $M_{t} = e^{tF}$ , recovering the scalar Feynman-Kac exponential when $F = - V$ is a potential. Identify where commutativity is used.

Solution

When $F$ is covariantly constant and commutes with all transports, $/ /_{t}^{- 1} F / /_{t} = F$ for every $t$ , so the defining equation becomes $d M_{t} / d t = M_{t} F = F M_{t}$ with $M_{0} = id$ , an autonomous linear ordinary differential equation whose solution is $M_{t} = e^{tF}$ . Commutativity is what lets us pull $F$ outside the path-ordering and write a single exponential rather than a time-ordered product; without it $M_{t}$ is the Dyson series $exp \int_{0}^{t} / /_{s}^{- 1} F / /_{s} d s$ . Setting $F = - V$ (a multiplication operator, automatically transport-commuting on a line bundle) gives the scalar weight $exp (- \int_{0}^{t} V (X_{s}) d s)$ of the Feynman-Kac formula.

Exercise 2 (medium, computation). On Euclidean $R^{n}$ , verify Varadhan's law directly from the explicit kernel $p_{t} (x, y) = (2 π t)^{- n /2} e^{- ∣ x - y ∣^{2} / (2 t)}$ , and explain why the prefactor $(2 π t)^{- n /2}$ does not affect the limit of $2 t lo g p_{t}$ .

Solution

Compute $2 t lo g p_{t} (x, y) = 2 t (- \frac{n}{2} lo g (2 π t)) - ∣ x - y ∣^{2} = - n t lo g (2 π t) - ∣ x - y ∣^{2}$ . The first term is $- n t lo g (2 π t)$ , which tends to $0$ as $t \to 0$ because $t lo g t \to 0$ . What survives is $- ∣ x - y ∣^{2} = - d (x, y)^{2}$ , the Euclidean distance squared. The prefactor contributes only the vanishing $t lo g t$ piece: the leading short-time behaviour of the kernel is governed by the Gaussian exponent, and any polynomial-in- $t$ prefactor washes out after multiplying the logarithm by $t$ . On a general manifold the prefactor becomes the Minakshisundaram-Pleijel expansion, but it still vanishes under the same scaling.

Exercise 3 (hard, proof). Using the Lichnerowicz formula $D^{2} = \nabla^{*} \nabla + \frac{1}{4} Scal$ for the pure spinor Dirac operator, write the Feynman-Kac-Ito representation of $e^{- t D^{2} /2}$ and identify the multiplicative functional explicitly. State which sign convention you adopt for $D^{2}$ versus $Δ$ .

Solution

Lichnerowicz gives $D^{2} = \nabla^{*} \nabla + \frac{1}{4} Scal = - Δ^{S} + \frac{1}{4} Scal$ with $Δ^{S}$ the spinor connection Laplacian. Then $- \frac{1}{2} D^{2} = \frac{1}{2} Δ^{S} - \frac{1}{8} Scal$ , so applying the key theorem with $F = - \frac{1}{4} Scal$ (after the factor-of-two bookkeeping) yields $$ \bigl(e^{-tD^2/2}\psi\bigr)(x) = \mathbb{E}_x!\left[ M_t, /!/_t^{-1}, \psi(X_t)\right], \qquad \frac{dM_t}{dt} = -\tfrac18, \mathrm{Scal}(X_t), M_t, $$ so $M_{t} = exp (- \frac{1}{8} \int_{0}^{t} Scal (X_{s}) d s)$ , a scalar because $Scal$ is a function commuting with transport. For a twisted Dirac operator $D_{E}$ on $S \otimes E$ the curvature endomorphism $F^{E / S}$ is added and $M_{t}$ becomes a genuine path-ordered exponential in $End (E)$ . I take $Δ^{S} \leq 0$ (geometer's sign) so $- Δ^{S} = \nabla^{*} \nabla \geq 0$ matches $D^{2} \geq 0$ .

Advanced results Master

Bismut's Brownian-bridge construction (1984). Bismut's probabilistic proof of the index theorem starts from the McKean-Singer identity $ind (D) = Str (e^{- t D^{2}})$ , valid for every $t > 0$ , and computes the right side as $t \to 0$ by probability. By the Feynman-Kac-Ito formula the supertrace is an integral over the diagonal of the heat kernel,

Str (e^{- t D^{2}}) = \int_{M} str_{x} (k_{t} (x, x)) d vol (x),

and the on-diagonal kernel is a Brownian-bridge expectation: condition the Brownian motion to return, $X_{t} = x$ , and average the multiplicative functional $M_{t}$ against the bridge measure $P_{x, x}^{t}$ scaled by the scalar heat kernel $p_{t} (x, x)$ . Concretely $k_{t} (x, x) = p_{t} (x, x) E_{x, x}^{t} [/ /_{t}^{- 1} M_{t}]$ , where the endpoint conditioning is the probabilistic analogue of evaluating on the diagonal.

As $t \to 0$ the bridge collapses onto the constant loop at $x$ , and the rescaled bridge converges to a Gaussian (Ornstein-Uhlenbeck) process governed by the Riemann curvature at $x$ — this is the probabilistic mechanism behind Mehler's formula. The Clifford-module curvature $F^{E / S}$ and the scalar curvature term enter $M_{t}$ linearly in the limit, the $p_{t} (x, x) \sim (2 π t)^{- n /2}$ prefactor supplies the correct power of $t$ , and the supertrace's Clifford-algebra bookkeeping (the Berezin/Patodi cancellation) extracts exactly the top-degree component. The surviving small- $t$ density is

t \to 0 lim str_{x} (k_{t} (x, x)) d vol = [A (M) \land ch (E)]_{top},

the Atiyah-Singer integrand, recovered without any Getzler coordinate rescaling.

Hypoellipticity and smoothness (Malliavin). That $p_{t} (x, y)$ is smooth even when the diffusion is degenerate is a theorem of Malliavin calculus: Hormander's bracket (bracket-generating) condition on the generating vector fields forces the Malliavin covariance matrix to be non-degenerate, hence the law of $X_{t}$ has a smooth density. For Brownian motion on a Riemannian manifold the horizontal vector fields together with one bracket span the tangent space, so the condition holds and $p_{t}$ is smooth off $t = 0$ .

Synthesis. Bismut's Brownian-bridge proof is the foundational reason the index theorem can be read as a statement about random paths: the supertrace of the heat operator localises because conditioned Brownian loops shrink to points, and the curvature they accumulate in the limit is exactly the $A$ -genus density. This is exactly the probabilistic mirror of the Getzler rescaling of 03.09.20, where coordinate dilation plays the role that path conditioning plays here; putting these together, the two proofs compute the same local index density by dual mechanisms, and the bridge is the recognition that Mehler's formula for the harmonic oscillator is simultaneously the small-time limit of the rescaled Getzler symbol and the Gaussian limit of the rescaled Brownian bridge. The central insight is that the Lichnerowicz curvature term $\frac{1}{4} Scal + F^{E / S}$ of 03.09.08, the scalar Feynman-Kac weight 02.15.04, and the manifold Brownian motion 03.02.45 are not three separate ingredients but one multiplicative functional carried along a random path, and this same functional generalises to the family setting where the Bismut superconnection of 03.09.23 organises the parametric heat kernel. The whole construction is dual to the analytic localisation: where Getzler rescales the symbol, Bismut conditions the measure, and the agreement of their limits is the central insight that probability and analysis are computing one geometric object.

Full proof set Master

Proposition (McKean-Singer constancy of the supertrace). Let $D$ be a self-adjoint Dirac operator on a $Z /2$ -graded Clifford module, with $D^{2}$ having discrete spectrum. Then $Str (e^{- t D^{2}})$ is independent of $t > 0$ and equals $ind (D) = dim ker D^{+} - dim ker D^{-}$ .

Proof. Write $D = D^{+} + D^{-}$ with $D^{\pm} : Γ (E^{\pm}) \to Γ (E^{\mp})$ , so $D^{2}$ preserves the grading and $D^{2} ∣_{E^{+}} = D^{-} D^{+}$ , $D^{2} ∣_{E^{-}} = D^{+} D^{-}$ . For each eigenvalue $λ > 0$ of $D^{2}$ , the operator $D$ restricts to an isomorphism between the $λ$ -eigenspaces $V_{λ}^{+} \subset Γ (E^{+})$ and $V_{λ}^{-} \subset Γ (E^{-})$ : if $D^{+} u = v$ with $u \in V_{λ}^{+}$ then $D^{-} v = D^{-} D^{+} u = λ u$ , and $\frac{1}{λ} D^{+}$ has inverse $\frac{1}{λ} D^{-}$ on these spaces. Hence $dim V_{λ}^{+} = dim V_{λ}^{-}$ for every $λ > 0$ . Now expand the supertrace over eigenspaces: $$ \mathrm{Str}(e^{-tD^2}) = \sum_{\lambda \geq 0} e^{-t\lambda}\bigl(\dim V_\lambda^+ - \dim V_\lambda^-\bigr). $$ Every $λ > 0$ term vanishes by the dimension equality just shown, leaving only $λ = 0$ : $dim ker D^{+} - dim ker D^{-}$ , which is $ind (D)$ and carries no $t$ . So the supertrace is constant in $t$ and equal to the index. $□$

Corollary (the short-time route is legitimate). Because $Str (e^{- t D^{2}})$ does not depend on $t$ , its value may be computed in the limit $t \to 0^{+}$ . The probabilistic content of Bismut's theorem is the evaluation of that limit as the Brownian-bridge integral above; the corollary is what licenses replacing the global spectral quantity (an integer) by a purely local small-time density. The same constancy underwrites the Getzler proof of 03.09.20, which is why the two methods, analytic and probabilistic, must agree.

Connections Master

The analytic counterpart of this unit is the heat-kernel proof of the index theorem in 03.09.20. There the small- $t$ limit is reached by Getzler's rescaling of the Clifford symbol; here it is reached by conditioning Brownian paths into bridges. Both lean on the McKean-Singer constancy proved above, and both produce the identical $A \land ch$ integrand — the agreement of a coordinate rescaling with a measure conditioning is the bridge between analysis and probability.
The family-index refinement lives in 03.09.23, the Bismut superconnection. The multiplicative functional carried along a single Brownian path generalises to a fibrewise heat kernel organised by the superconnection over a parameter space; Bismut's 1986 family-index proof is the parametric heat-equation cousin of his 1984 probabilistic proof, and the curvature term $F^{E / S}$ appearing in $M_{t}$ here is the fibre-direction piece of the superconnection curvature there.
The Lichnerowicz decomposition $D^{2} = \nabla^{*} \nabla + \frac{1}{4} Scal + F^{E / S}$ from 03.09.08 is what the multiplicative functional encodes pathwise: the connection Laplacian becomes Brownian motion with stochastic parallel transport, and the zeroth-order curvature endomorphism becomes the path-ordered exponential $M_{t}$ . The probabilistic heat kernel is the stochastic reading of Lichnerowicz.
The diffusion itself, Brownian motion on a manifold via the Eells-Elworthy-Malliavin horizontal lift, is constructed in the co-produced 03.02.45; the scalar weighting mechanism, the Feynman-Kac formula for a Schrodinger semigroup, is developed in the co-produced 02.15.04. This unit fuses the two into the bundle-valued Feynman-Kac-Ito formula that Bismut needed for spinors.

Historical & philosophical context Master

The idea that heat flow and random motion are the same dates to Einstein's 1905 account of Brownian motion and to Wiener's 1923 rigorous construction of the path measure, but its decisive turn for geometry came with Kac's 1949 reformulation of the Feynman path integral as an honest probabilistic average, the Feynman-Kac formula ^{[Kac 1949]}. Varadhan's 1967 theorem that $2 t lo g p_{t} \to - d^{2}$ revealed that the heat kernel encodes the Riemannian distance, fusing analysis and geometry through probability ^{[Varadhan 1967]}. The frame-bundle construction of manifold Brownian motion is due to Eells-Elworthy and Malliavin, the latter inventing in 1976 a stochastic calculus of variations precisely to give a probabilistic proof of Hormander's hypoellipticity theorem ^{[Malliavin 1976]}.

The synthesis was Bismut's. In two 1984 papers he gave a fully probabilistic proof of the Atiyah-Singer index theorem and of the Lefschetz fixed-point formulas, representing the heat kernel of a Dirac operator by the Brownian bridge together with stochastic parallel transport and the Ito-Feynman-Kac functional carrying the Clifford-module curvature ^{[Bismut 1984]}. Philosophically the achievement is a statement about where mathematical content lives: an integer invariant of an elliptic operator, defined by global spectral data, is recovered as the short-time limit of an average over random loops that shrink to points. The localisation that Getzler engineered by rescaling coordinates, Bismut obtained by letting probability concentrate measure on constant paths — two faces of the principle that the index is local. That a careful averaging over jagged, nowhere-differentiable Brownian paths reproduces the smooth characteristic-class integrand of Atiyah-Singer is among the more striking demonstrations that probability is a legitimate organ of geometry, not merely a computational convenience.

Bibliography Master

@article{Bismut1984I,
  author  = {Bismut, Jean-Michel},
  title   = {The {Atiyah-Singer} theorems: a probabilistic approach. {I}. The index theorem},
  journal = {Journal of Functional Analysis},
  volume  = {57},
  number  = {1},
  pages   = {56--99},
  year    = {1984}
}

@article{Bismut1984II,
  author  = {Bismut, Jean-Michel},
  title   = {The {Atiyah-Singer} theorems: a probabilistic approach. {II}. The {Lefschetz} fixed point formulas},
  journal = {Journal of Functional Analysis},
  volume  = {57},
  number  = {3},
  pages   = {329--348},
  year    = {1984}
}

@book{BGV1992,
  author    = {Berline, Nicole and Getzler, Ezra and Vergne, Mich\`ele},
  title     = {Heat Kernels and Dirac Operators},
  series    = {Grundlehren der mathematischen Wissenschaften},
  volume    = {298},
  publisher = {Springer},
  year      = {1992},
  note      = {Ch. 11, the probabilistic / Mehler-formula approach}
}

@book{Hsu2002,
  author    = {Hsu, Elton P.},
  title     = {Stochastic Analysis on Manifolds},
  series    = {Graduate Studies in Mathematics},
  volume    = {38},
  publisher = {American Mathematical Society},
  year      = {2002}
}

@article{Varadhan1967,
  author  = {Varadhan, S. R. S.},
  title   = {On the behavior of the fundamental solution of the heat equation with variable coefficients},
  journal = {Communications on Pure and Applied Mathematics},
  volume  = {20},
  pages   = {431--455},
  year    = {1967}
}

@incollection{Malliavin1976,
  author    = {Malliavin, Paul},
  title     = {Stochastic calculus of variation and hypoelliptic operators},
  booktitle = {Proceedings of the International Symposium on Stochastic Differential Equations (Kyoto 1976)},
  pages     = {195--263},
  publisher = {Wiley},
  year      = {1978}
}

@article{Kac1949,
  author  = {Kac, Mark},
  title   = {On distributions of certain {Wiener} functionals},
  journal = {Transactions of the American Mathematical Society},
  volume  = {65},
  pages   = {1--13},
  year    = {1949}
}

Prerequisites

03.09.20
03.09.23
03.09.08

Tier anchors

beginner: A random walker explores a curved surface; where it tends to be after a short time secretly measures the straight-line distance, and Bismut taught the walker to carry a spinning gyroscope
intermediate: Berline-Getzler-Vergne Ch. 11 (Mehler's formula, the probabilistic approach); Hsu *Stochastic Analysis on Manifolds* Ch. 4-5; Bismut 1984 *The Atiyah-Singer theorems: a probabilistic approach* (J. Funct. Anal. 57) §1-§2
master: Bismut 1984 *The Atiyah-Singer theorems: a probabilistic approach I, II* (J. Funct. Anal. 57); Berline-Getzler-Vergne *Heat Kernels and Dirac Operators* (Grundlehren 298) Ch. 11; Hsu *Stochastic Analysis on Manifolds* (GSM 38); Varadhan 1967 *On the behavior of the fundamental solution of the heat equation with variable coefficients* (Comm. Pure Appl. Math. 20); Ikeda-Watanabe *Stochastic Differential Equations and Diffusion Processes* (North-Holland)

References

Bismut — The Atiyah-Singer theorems: a probabilistic approach. I. The index theorem · J. Funct. Anal. 57 (1984), 56-99 — originator paper
Bismut — The Atiyah-Singer theorems: a probabilistic approach. II. The Lefschetz fixed point formulas · J. Funct. Anal. 57 (1984), 329-348
Berline-Getzler-Vergne — Heat Kernels and Dirac Operators · Grundlehren 298, Springer 1992, Ch. 11 (the probabilistic / Mehler-formula approach)
Hsu — Stochastic Analysis on Manifolds · Graduate Studies in Mathematics 38, AMS 2002, Ch. 3-5 (Eells-Elworthy-Malliavin construction, Feynman-Kac)
Varadhan — On the behavior of the fundamental solution of the heat equation with variable coefficients · Comm. Pure Appl. Math. 20 (1967), 431-455 — short-time asymptotics
Ikeda-Watanabe — Stochastic Differential Equations and Diffusion Processes · North-Holland 1989, Ch. V (manifold-valued diffusions, stochastic parallel transport)
Malliavin — Stochastic calculus of variation and hypoelliptic operators · Proc. Intl. Symp. SDE Kyoto 1976, 195-263 — Malliavin calculus, Hörmander smoothness

Estimated time

beginner: 20m
intermediate: 48m
master: 85m