02.15.04 · analysis / stochastic-analysis

The Feynman-Kac formula

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Anchor (Master): Karatzas-Shreve Brownian Motion and Stochastic Calculus §§4.4, 5.7; Kac On distributions of certain Wiener functionals (1949); Simon Functional Integration and Quantum Physics Ch. 1-3 (Euclidean Schrödinger semigroups)

Intuition Beginner

Heat spreads. Drop a bead of dye into still water and watch it bloom outward, the sharp dot softening into a wider and wider smudge until the colour is spread thin everywhere. The mathematics of that blooming is the heat equation, and for a long time the only way to solve it was to treat the dye as a continuous field and grind through calculus on it.

There is a second picture, and it is the heart of this unit. Forget the field. Imagine the dye is made of millions of tiny specks, each one jittering at random, taking a haphazard walk with no memory and no goal. Each speck wanders off in its own direction, and the smudge you see is just the crowd of them spread out. The amount of colour at a spot tomorrow is the average over all the random wanders of where a speck happens to land.

That swap — from solving an equation about a smooth field to averaging over random walks — is the formula this unit is built around. It turns a question about heat into a question about the typical fate of a wanderer.

Visual Beginner

A starting dot sits at the centre. From it, several thin jagged paths spray outward, each one a random wiggling walk that ends somewhere different. Over each endpoint sits a faint bell-shaped hump showing how likely a walker is to finish there: tall near the start, low far away.

The bell shape is the punchline. If you release a huge swarm of walkers from one point and let them jitter for a fixed time, the crowd settles into a bell-shaped pile, fattest at the centre and thinning toward the edges. The height of the value we want at any spot is the average value the walkers carry to that spot. To find the temperature at a place and time, you do not solve the heat equation directly; you ask where the random walkers go and take the mean of what they bring.

Worked example Beginner

Start a single random walker at a point on a line and let it jitter for one unit of time. Where does it end up? Not at a fixed place, but spread out around the start in a bell-shaped pile. Suppose we want to know the value of some starting pattern, averaged over where the walker lands.

Take the starting pattern to be a smooth hill centred at zero. Release the walker from a point to the right of zero. Most of the walker's possible endpoints sit near where it started, on the right slope of the hill, but a good fraction drift left toward the peak and some drift further right down the far slope. The average of the hill's height over all those endpoints is the answer the formula gives for the value at that point after one unit of time.

Now release the walker from the very peak. Its endpoints spread symmetrically to both sides, all sampling lower parts of the hill, so the average comes out a little below the peak height. The pattern has slumped: the top is pulled down, the slopes filled in. That slumping is the dye blooming. We never solved an equation; we released walkers, looked at where they landed, and averaged the starting hill over those landing spots. The bloomed pattern is what the average produces.

Check your understanding Beginner

Exercise (easy, multiple-choice).

You want the value of a solution to the heat equation at a point $x$ after time $t$ . The path-averaging picture says to:

A. Solve the equation by hand and read off the value at $x$
B. Release random walkers from $x$ , let them jitter for time $t$ , and average the starting pattern over where they land
C. Add up the starting pattern at every point
D. Take the highest value the starting pattern ever reaches

Hint

The whole point is to replace solving the field equation with averaging over random wanderers started at the point of interest.

Answer

B. The value at $x$ after time $t$ is the average of the starting pattern evaluated at the random endpoints of walkers released from $x$ and run for time $t$ . Option A is the very thing we are avoiding; option C ignores where walkers actually go; option D confuses the average with the maximum.

Formal definition Intermediate+

Let $L$ be a second-order elliptic differential operator on $R^{n}$ , $$ L = \tfrac{1}{2} \sum_{i,j} a_{ij}(x), \partial_{x_i} \partial_{x_j} + \sum_i b_i(x), \partial_{x_i}, \qquad a(x) = \sigma(x)\sigma(x)^\top, $$ with $σ, b$ Lipschitz and of linear growth, so that the stochastic differential equation $d X_{s} = b (X_{s}) d s + σ (X_{s}) d B_{s}$ has a unique strong solution — the diffusion with generator $L$ , in the sense of the construction co-produced as 02.15.03. Write $X^{x}$ for the solution started at $X_{0} = x$ , write $P_{x}$ and $E_{x}$ for probability and expectation under that initial condition, and let $V : R^{n} \to R$ be bounded below and continuous (the potential, or killing rate). The expectation $E_{x}$ is the Lebesgue integral of 02.07.04 against the law of the path.

Definition (Feynman-Kac functional). Given a measurable $u_{0} : R^{n} \to R$ of at most polynomial growth, define $$ u(t,x) ;=; \mathbb{E}_x!\left[, u_0(X_t), \exp!\Big(-!\int_0^t V(X_s),ds\Big) \right], \qquad t \ge 0,\ x \in \mathbb{R}^n. $$ The exponential weight is the multiplicative functional $M_{t} = exp (- \int_{0}^{t} V (X_{s}) d s)$ ; it discounts a path by the accumulated potential it has traversed. When $V \equiv 0$ the weight is the constant $1$ and $u (t, x) = E_{x} [u_{0} (X_{t})]$ is a pure average of the initial data over the diffusion.

The pure heat case. Take $L = \frac{1}{2} Δ$ and $V \equiv 0$ . Then $X = B$ is Brownian motion, and $$ u(t,x) = \mathbb{E}x[u_0(B_t)] = \int{\mathbb{R}^n} u_0(y), p_t(x,y),dy, \qquad p_t(x,y) = (2\pi t)^{-n/2} \exp!\Big(-\frac{|x-y|^2}{2t}\Big). $$ The Brownian transition density $p_{t} (x, y)$ is exactly the Gaussian heat kernel of 02.13.03: the probabilistic object (the law of $B_{t}$ started at $x$ ) and the analytic object (the fundamental solution of $\partial_{t} = \frac{1}{2} Δ$ ) are one and the same function. This identity is the seed of the whole theory; everything below is the statement that the diffusion average solves the corresponding parabolic equation.

Counterexamples to common slips

The integral $\int_{0}^{t} V (X_{s}) d s$ is taken along the random path, not at the endpoint: it is a functional of the entire trajectory ${X_{s} : 0 \leq s \leq t}$ , and replacing it by $t V (X_{t})$ is wrong. Two paths with the same endpoint but different histories carry different weights.
Boundedness of $V$ below is what keeps $M_{t} \leq e^{- t i n f V} < \infty$ and the expectation finite. If $V$ is unbounded below, the exponential can blow up and the formula may diverge; the Schrödinger case with a deep attractive well is exactly where this care is needed.
The formula represents the solution of the backward Cauchy problem $\partial_{t} u = Lu - V u$ with initial data $u_{0}$ , not the forward Fokker-Planck (Kolmogorov forward) equation for densities. The generator $L$ acts in the space variable on $u$ ; its formal adjoint $L^{*}$ governs the evolution of densities instead.

Key theorem with proof Intermediate+

Theorem (Feynman-Kac). Let $u \in C^{1, 2} ((0, \infty) \times R^{n})$ be a bounded solution of the Cauchy problem $$ \partial_t u = L u - V u \quad (t>0), \qquad u(0,\cdot) = u_0, $$ with $V$ bounded below and continuous and $u_{0}$ continuous of polynomial growth. Then $u$ has the representation $$ u(t,x) ;=; \mathbb{E}_x!\left[, u_0(X_t), \exp!\Big(-!\int_0^t V(X_s),ds\Big) \right]. $$

Proof. Fix $t > 0$ and $x$ . For $0 \leq s \leq t$ consider the discounted process $$ Y_s ;=; u(t-s,, X_s), \exp!\Big(-!\int_0^s V(X_r),dr\Big) ;=; u(t-s, X_s), M_s. $$ We show $Y$ is a martingale under $P_{x}$ . Apply Itô's formula of 02.15.02 to the product $u (t - s, X_{s}) M_{s}$ , noting $d M_{s} = - V (X_{s}) M_{s} d s$ (the weight has finite variation, so it contributes no quadratic term and no cross-variation with $X$ ). Writing $\partial_{t} u$ for the time derivative and using that the generator of $X$ is $L$ , $$ dY_s = M_s\Big[, -\partial_t u + L u - V u ,\Big](t-s, X_s),ds ;+; M_s, \nabla u(t-s,X_s)^\top \sigma(X_s), dB_s. $$ The drift bracket is $- \partial_{t} u + (Lu - V u)$ ; by hypothesis $u$ satisfies $\partial_{t} u = Lu - V u$ , so the bracket vanishes identically. Hence $$ dY_s = M_s, \nabla u(t-s,X_s)^\top \sigma(X_s), dB_s, $$ a pure stochastic-integral term. Under the boundedness of $u$ and the growth control on $σ$ and $\nabla u$ , the integrand lies in the class for which the Itô integral is a genuine martingale (localise by a stopping sequence and pass to the limit using dominated convergence, the weight $M_{s} \leq e^{- s i n f V}$ supplying the domination). Therefore $E_{x} [Y_{t}] = E_{x} [Y_{0}]$ . Now evaluate the endpoints: at $s = 0$ , $Y_{0} = u (t, X_{0}) M_{0} = u (t, x)$ (a constant); at $s = t$ , $Y_{t} = u (0, X_{t}) M_{t} = u_{0} (X_{t}) exp (- \int_{0}^{t} V (X_{r}) d r)$ . Equating the two expectations gives $$ u(t,x) = \mathbb{E}_x!\left[ u_0(X_t)\exp!\Big(-!\int_0^t V(X_r),dr\Big)\right]. \qquad \square $$

Bridge. This martingale argument builds toward every later use of stochastic representation, and it appears again in the killed-diffusion and Dirichlet-problem refinements of the Master tier, where a stopping time replaces the fixed horizon $t$ . The foundational reason the proof works is that the diffusion's generator $L$ is defined so that $u (t - s, X_{s})$ has drift $Lu$ under Itô's formula; the discount $M_{s}$ is engineered so that its $- V u$ drift cancels exactly the $- V u$ term in the equation, and this is exactly why the potential enters as a path integral in the exponent rather than as a pointwise factor. The construction generalises the pure-heat identity $u (t, x) = E_{x} [u_{0} (B_{t})]$ — there $V \equiv 0$ , $M_{s} \equiv 1$ , and $Y_{s} = u (t - s, B_{s})$ is already a martingale because $u$ solves $\partial_{t} u = \frac{1}{2} Δ u$ — to an arbitrary elliptic $L$ with potential. The central insight is that the Cauchy problem and the path average are two readings of one martingale, and putting these together the heat kernel of 02.13.03 is recovered as the Brownian transition density. The bridge is that solving a parabolic PDE and computing an expectation over diffusion paths are the same operation seen from two sides.

Exercises Intermediate+

Exercise 1 (easy, short-answer).

Specialise the Feynman-Kac theorem to $L = \frac{1}{2} Δ$ , $V \equiv 0$ , and write the solution as an integral against the Gaussian kernel. Identify the kernel with the heat kernel of 02.13.03.

Hint

With $V \equiv 0$ the weight is $1$ and $X = B$ . The law of $B_{t}$ started at $x$ has an explicit Gaussian density.

Answer

The weight $M_{t} = 1$ , so $u (t, x) = E_{x} [u_{0} (B_{t})]$ . Since $B_{t}$ under $P_{x}$ is normal with mean $x$ and covariance $t I$ , its density is $p_{t} (x, y) = (2 π t)^{- n /2} exp (- ∣ x - y ∣^{2} /2 t)$ , and $u (t, x) = \int u_{0} (y) p_{t} (x, y) d y$ . This $p_{t}$ is precisely the fundamental solution of $\partial_{t} u = \frac{1}{2} Δ u$ from 02.13.03; the probabilistic transition density and the analytic heat kernel coincide. Rubric: full credit for the convolution form and the explicit identification with the heat kernel.

Exercise 3 (medium, short-answer).

Explain precisely where, in the martingale proof, the hypothesis that $u$ solves $\partial_{t} u = Lu - V u$ is used, and what would survive if instead $\partial_{t} u = Lu - V u + g$ for a source $g$ .

Hint

The drift bracket of $d Y_{s}$ is $- \partial_{t} u + Lu - V u$ . Track what it becomes when a source $g$ is present.

Answer

The equation is used exactly at the step where the drift bracket $M_{s} [- \partial_{t} u + Lu - V u] (t - s, X_{s})$ is set to zero; without the PDE, $Y_{s}$ would not be a martingale and the endpoint identity would fail. If instead $\partial_{t} u = Lu - V u + g$ , the bracket equals $- g$ , so $d Y_{s} = - M_{s} g (t - s, X_{s}) d s + (martingale)$ . Taking expectations and integrating the drift gives the inhomogeneous (Duhamel) form $u (t, x) = E_{x} [u_{0} (X_{t}) M_{t}] + E_{x} [\int_{0}^{t} M_{s} g (t - s, X_{s}) d s]$ , the probabilistic counterpart of Duhamel's principle from 02.13.03. Rubric: full credit for locating the cancellation and producing the Duhamel correction term.

Exercise 4 (hard, short-answer).

State and justify the Feynman-Kac representation for the elliptic (time-independent) Dirichlet problem $Lu = V u$ on a bounded domain $D$ with $u = f$ on $\partial D$ , using the first exit time $τ_{D} = in f {s : X_{s} \in / D}$ .

Hint

Apply the same Itô computation to $u (X_{s}) M_{s}$ (no time argument now) and stop at $τ_{D}$ . The optional-stopping theorem replaces the fixed horizon.

Answer

Set $Y_{s} = u (X_{s}) exp (- \int_{0}^{s} V (X_{r}) d r)$ . Itô's formula gives $d Y_{s} = M_{s} [Lu - V u] (X_{s}) d s + (martingale)$ ; since $Lu = V u$ in $D$ , the drift vanishes for $s < τ_{D}$ , so $Y_{s \land τ_{D}}$ is a martingale. By optional stopping (valid because $E_{x} [τ_{D}] < \infty$ for a uniformly elliptic diffusion on a bounded domain, and $M \leq 1$ when $V \geq 0$ ), $E_{x} [Y_{τ_{D}}] = Y_{0} = u (x)$ . At $τ_{D}$ the diffusion sits on $\partial D$ where $u = f$ , giving $$ u(x) = \mathbb{E}x!\left[ f(X{\tau_D})\exp!\Big(-!\int_0^{\tau_D} V(X_r),dr\Big)\right]. $$ The exponential weight is the survival probability of a particle killed at rate $V$ before it reaches the boundary. Rubric: full credit for the stopped martingale, the optional-stopping justification, and the boundary evaluation.

Exercise 5 (hard, short-answer).

Derive Kac's occupation-time formula: for $V = λ 1_{A}$ the indicator of a set $A$ , show $E_{x} [exp (- λ T_{t}^{A})]$ with $T_{t}^{A} = \int_{0}^{t} 1_{A} (X_{s}) d s$ the time spent in $A$ , and relate it to the formula.

Hint

Substitute $V = λ 1_{A}$ into the Feynman-Kac functional with $u_{0} \equiv 1$ and read off what the path integral in the exponent measures.

Answer

With $u_{0} \equiv 1$ and $V = λ 1_{A}$ , the exponent is $- \int_{0}^{t} λ 1_{A} (X_{s}) d s = - λ T_{t}^{A}$ , where $T_{t}^{A}$ is the occupation time of $A$ up to $t$ . Hence $u (t, x) = E_{x} [exp (- λ T_{t}^{A})]$ , the Laplace transform (in $λ$ ) of the occupation-time distribution. By the theorem this same $u$ solves $\partial_{t} u = Lu - λ 1_{A} u$ , so the generating function of the time a diffusion spends in $A$ is governed by a parabolic equation with a potential supported on $A$ . This is the identity Kac introduced in 1949 to compute the distribution of $\int_{0}^{t} 1_{A} (B_{s}) d s$ for Brownian motion. Rubric: full credit for identifying the occupation time and the Laplace-transform reading.

Advanced results Master

The killed diffusion and the probabilistic semigroup. The potential $V \geq 0$ admits a literal probabilistic reading: enlarge the state space by a cemetery point $\partial$ and kill the path $X$ at rate $V (X_{s})$ , i.e. send it to $\partial$ over $[s, s + d s]$ with probability $V (X_{s}) d s$ . The survival probability up to time $t$ , conditional on the trajectory, is exactly $exp (- \int_{0}^{t} V (X_{s}) d s)$ , so the Feynman-Kac functional $u (t, x) = E_{x} [u_{0} (X_{t})]$ for the killed diffusion $X$ (with $u_{0} (\partial) := 0$ ). The family $P_{t}^{V} u_{0} := u (t, \cdot)$ is then a strongly continuous semigroup of bounded operators, the Schrödinger semigroup, with infinitesimal generator $L - V$ . The Dirichlet problem of Exercise 4 is the same construction with killing replaced by absorption at the boundary $\partial D$ ; killed and absorbed diffusions are the two faces of a sub-Markovian semigroup, and the heat content of a domain — the total surviving mass — is the Feynman-Kac average of the constant $1$ .

The Euclidean bridge to quantum mechanics. Place $L = \frac{1}{2} Δ$ and read $H = - \frac{1}{2} Δ + V$ as a Schrödinger Hamiltonian. The unitary quantum evolution $e^{- i t H}$ of the real-time Schrödinger equation $i \partial_{t} ψ = H ψ$ has, formally, the same generator as the Feynman-Kac semigroup $e^{- t H}$ after the substitution $t \mapsto i t$ — the Wick rotation to imaginary (Euclidean) time. Feynman's 1948 sum-over-histories wrote $e^{- i t H}$ as an oscillatory integral over paths weighted by $e^{i S}$ , an object with no underlying measure. Kac's contribution was to observe that in imaginary time the oscillatory weight becomes the genuine, positive Wiener weight, so $e^{- t H}$ acquires a rigorous path-integral kernel $$ \big(e^{-tH}\big)(x,y) = \mathbb{E}_{x\to y}!\left[\exp!\Big(-!\int_0^t V(B_s),ds\Big)\right] p_t(x,y), $$ the expectation taken over the Brownian bridge from $x$ to $y$ . This is the precise sense in which probability furnishes a mathematically honest path integral while the physicists' real-time version remains formal; it cross-links the construction of Brownian motion and the path integral in 08.14.01, where the Wiener measure is built as the measure the present formula integrates against.

Spectral consequences. Because $P_{t}^{V} = e^{- t (L + V)}$ on $L^{2}$ is a positivity-preserving semigroup with a Gaussian-dominated kernel, the path-integral representation gives quantitative control of the spectrum of $H = - \frac{1}{2} Δ + V$ : lower bounds on the kernel yield upper bounds on eigenvalues, ground-state positivity follows from the strict positivity of the Wiener weight, and large- $t$ asymptotics of $tr e^{- t H}$ feed Tauberian estimates of the eigenvalue-counting function. The functional-integral method of Simon's monograph turns each analytic question about $H$ into a question about a Brownian functional, and the probabilistic positivity of the weight is what makes the bridge so productive.

Synthesis. The Feynman-Kac formula is the foundational reason the heat kernel of 02.13.03 is the Brownian transition density and not merely similar to it: solving the Cauchy problem and averaging over diffusion paths are the same martingale read two ways, and this is exactly the duality between the generator $L$ acting on functions and the diffusion $X$ realising it. The central insight is that the potential enters not pointwise but as a path integral $\int_{0}^{t} V (X_{s}) d s$ in an exponent, because the multiplicative functional that discounts a path is precisely the survival probability of a particle killed at rate $V$ ; this generalises the pure-heat average to the full Schrödinger semigroup $e^{- t (L + V)}$ and is dual, under Wick rotation $t \mapsto i t$ , to the formal real-time path integral of quantum mechanics. Putting these together, the killed diffusion, the Dirichlet problem, Kac's occupation-time identity, and the Euclidean quantum propagator are all one construction specialised: a stopped or weighted average over the paths of a diffusion. The bridge is that the analytic semigroup $e^{- t H}$ and the probabilistic average over Wiener paths are two names for the operator this unit builds, and that operator appears again in 08.14.01 as the rigorous content of the path integral and builds toward the spectral theory of Schrödinger operators.

Full proof set Master

The martingale verification of the parabolic Feynman-Kac formula is given in full in the Key theorem section. The remaining Master claims are recorded here.

Proposition (semigroup property of the Feynman-Kac propagator). For $V$ bounded below, the operators $P_{t}^{V} u_{0} (x) = E_{x} [u_{0} (X_{t}) exp (- \int_{0}^{t} V (X_{s}) d s)]$ form a semigroup: $P_{t + r}^{V} = P_{t}^{V} \circ P_{r}^{V}$ for $t, r \geq 0$ .

Proof. Fix $t, r \geq 0$ and a bounded measurable $u_{0}$ . By the Markov property of the diffusion $X$ (the defining property of the construction in 02.15.03), conditioning on $F_{t}$ and using that $X$ restarts from $X_{t}$ with a fresh independent driving Brownian motion, $$ \mathbb{E}x!\Big[ u_0(X{t+r}), e^{-\int_0^{t+r} V(X_s),ds} ,\Big|, \mathcal F_t \Big] = e^{-\int_0^t V(X_s),ds}; \mathbb{E}_{X_t}!\Big[ u_0(X_r), e^{-\int_0^r V(X_s),ds}\Big], $$ where the last factor is $P_{r}^{V} u_{0} (X_{t})$ by the strong Markov restart. The accumulated weight splits as $\int_{0}^{t + r} = \int_{0}^{t} + \int_{t}^{t + r}$ , and the second piece is measured by the restarted path, which is the source of the factorisation. Taking $E_{x}$ of both sides and using the tower property, $$ P_{t+r}^V u_0(x) = \mathbb{E}_x!\Big[ e^{-\int_0^t V(X_s),ds}, P_r^V u_0(X_t)\Big] = P_t^V\big(P_r^V u_0\big)(x). \qquad \square $$

Proposition (Kac moment formula for Brownian occupation). Let $B$ be Brownian motion on $R$ and $V \geq 0$ bounded continuous. Then $w (x) := E_{x} [\int_{0}^{\infty} V (B_{s}) e^{- \int_{0}^{s} V (B_{r}) d r} d s]$ — the expected total accumulated killing — equals $1$ wherever the diffusion is certain to be eventually killed, and more generally $v (x) := E_{x} [exp (- \int_{0}^{\infty} V (B_{s}) d s)]$ solves $\frac{1}{2} v^{''} = V v$ .

Proof. For the second claim, apply the Itô computation of the Key theorem to $Y_{s} = v (B_{s}) exp (- \int_{0}^{s} V (B_{r}) d r)$ with the time-independent $v$ . Itô's formula gives $d Y_{s} = M_{s} (\frac{1}{2} v^{''} - V v) (B_{s}) d s + M_{s} v^{'} (B_{s}) d B_{s}$ . For $Y$ to be the martingale that makes $E_{x} [Y_{\infty}] = Y_{0} = v (x)$ consistent with the definition $v (x) = E_{x} [M_{\infty}]$ (here $u_{0} \equiv 1$ so $Y_{\infty} = M_{\infty}$ ), the drift must vanish, i.e. $\frac{1}{2} v^{''} = V v$ . Conversely a bounded solution of that elliptic equation yields, by the stopped-martingale argument of Exercise 4 with $τ_{D} ↑ \infty$ , the representation $v (x) = E_{x} [M_{\infty}]$ . For the first claim, integrate the identity $\frac{d}{d s} E_{x} [M_{s}] = - E_{x} [V (B_{s}) M_{s}]$ from $0$ to $\infty$ : the left side telescopes to $E_{x} [M_{\infty}] - 1 = v (x) - 1$ , so $w (x) = 1 - v (x)$ , which is $1$ exactly when $v (x) = 0$ , i.e. when killing is certain. $□$

Proposition (Euclidean kernel via the Brownian bridge). For $L = \frac{1}{2} Δ$ and $V$ bounded below, the Schrödinger semigroup $e^{- t H}$ , $H = - \frac{1}{2} Δ + V$ , has integral kernel $K_{t} (x, y) = p_{t} (x, y) E_{x \to y}^{(t)} [exp (- \int_{0}^{t} V (B_{s}) d s)]$ , the inner expectation over the Brownian bridge from $x$ to $y$ in time $t$ .

Proof. Disintegrate the Wiener measure started at $x$ over the endpoint $B_{t} = y$ : the law of $B_{t}$ has density $p_{t} (x, \cdot)$ , and conditionally on $B_{t} = y$ the path is a Brownian bridge $B^{x \to y}$ . Hence for bounded $u_{0}$ , $$ P_t^V u_0(x) = \mathbb{E}x\big[u_0(B_t)M_t\big] = \int{\mathbb{R}^n} u_0(y), p_t(x,y), \mathbb{E}{x\to y}^{(t)}!\big[M_t\big],dy = \int u_0(y),K_t(x,y),dy, $$ by Fubini (justified since $V$ is bounded below, so $M_{t} \leq e^{- t i n f V}$ and the integrand is dominated by $∣ u_{0} (y) ∣ e^{- t i n f V} p_{t} (x, y)$ , integrable for polynomially bounded $u_{0}$ , invoking the dominated convergence of 02.07.04). Reading off the kernel of $P_{t}^{V} = e^{- t H}$ gives $K_t(x,y) = p_t(x,y),\mathbb{E}{x\to y}^{(t)}[M_t] $.$ \square$

Connections Master

The Itô integral and Itô's formula 02.15.02 supply the engine of every proof in this unit. Itô's formula applied to the discounted process $u (t - s, X_{s}) M_{s}$ is what produces the drift bracket $- \partial_{t} u + Lu - V u$ ; the cancellation of that bracket against the Cauchy problem is the entire mechanism of the representation, and the stochastic-integral term it leaves behind is the martingale whose constancy of expectation yields the formula.

The diffusion attached to a generator and the SDE construction 02.15.03, co-produced in this wave, is what gives meaning to "the diffusion $X$ with generator $L$ " used throughout. That unit builds $X$ as the strong solution of $d X = b d t + σ d B$ and identifies its generator as $L$ ; the present unit consumes that identification at the step where Itô's formula turns $u (X_{s})$ into a process with drift $Lu$ . The Markov and strong-Markov properties established there are exactly what the semigroup proposition here invokes. The two units are the forward and backward faces of one object: 02.15.03 runs the paths, this unit averages over them.

The heat equation, heat kernel, and Duhamel principle 02.13.03 is the analytic prerequisite and the destination. The Gaussian heat kernel constructed there as the fundamental solution of $\partial_{t} = \frac{1}{2} Δ$ is recovered here as the Brownian transition density, and Duhamel's principle for the inhomogeneous equation reappears (Exercise 3) as the probabilistic source term $E_{x} [\int_{0}^{t} M_{s} g d s]$ . The maximum principle of the parabolic theory is mirrored by the positivity of the diffusion average.

The Lebesgue integral and monotone convergence 02.07.04 is what the expectation $E_{x}$ is: the integral of the path functional against the law of the diffusion. The dominated- and monotone-convergence theorems of that unit are invoked at each localisation step where a stopped martingale is passed to its limit and where Fubini disintegrates the Wiener measure over the endpoint.

Brownian motion, the Wiener measure, and the path integral 08.14.01 is the physics-facing counterpart. There the Wiener measure is constructed as the measure on path space that the Feynman-Kac functional integrates against, and the formal real-time Feynman path integral is presented; this unit supplies the rigorous imaginary-time version of that object, the Euclidean propagator $e^{- t H}$ , and the Wick rotation $t \mapsto i t$ is the precise dictionary between the two.

Historical & philosophical context Master

Richard Feynman's 1948 Space-time approach to non-relativistic quantum mechanics (Reviews of Modern Physics 20, 367–387) ^{[Feynman 1948]} proposed that the quantum propagator $e^{- i t H}$ be written as a sum over all paths connecting two points, each weighted by $e^{i S}$ with $S$ the classical action. The proposal was physically luminous and analytically suspect: the oscillatory weight $e^{i S}$ defines no countably additive measure on path space, so the "integral" was a formal limit of finite-dimensional integrals with no underlying measure-theoretic content. Mark Kac, who had attended Feynman's lectures, recognised that the difficulty evaporates in imaginary time. His 1949 paper On distributions of certain Wiener functionals (Transactions of the AMS 65, 1–13) ^{[Kac 1949]} established the now-standard identity by computing the distribution of $\int_{0}^{t} 1_{A} (B_{s}) d s$ , the occupation time, and recognising the exponential functional $E_{x} [e^{- \int_{0}^{t} V (B_{s}) d s}]$ as the solution of a heat equation with potential. Where Feynman summed $e^{i S}$ over real-time paths, Kac integrated the genuine, positive Wiener weight over Brownian paths — the same formula with the oscillation traded for decay.

The philosophical payoff is a lesson about which structures are real. The real-time path integral remains, to this day, a formal device that physicists wield with great success but mathematicians cannot ground as a measure; the imaginary-time version is a theorem about the Wiener measure, complete with a martingale proof. The mathematical honesty of the construction is bought entirely by the positivity of the Brownian weight, which is what lets the Wiener measure exist as a probability measure in the first place. The standard modern accounts — Øksendal's Stochastic Differential Equations (2003) ^{[Øksendal 2003]}, Karatzas and Shreve's Brownian Motion and Stochastic Calculus (1991) ^{[Karatzas-Shreve 1991]}, and Barry Simon's Functional Integration and Quantum Physics (1979) ^{[Simon 1979]} — treat the formula not as a curiosity but as the central conduit between analysis and probability: every spectral question about a Schrödinger operator becomes a question about a Brownian functional, and the heat kernel of partial differential equations becomes the transition law of a stochastic process. The recurring theme is that the right way to compute with a generator is sometimes to run the process it generates.

Bibliography Master

@article{kac1949,
  author  = {Kac, Mark},
  title   = {On distributions of certain {W}iener functionals},
  journal = {Transactions of the American Mathematical Society},
  volume  = {65},
  number  = {1},
  pages   = {1--13},
  year    = {1949}
}

@article{feynman1948,
  author  = {Feynman, Richard P.},
  title   = {Space-time approach to non-relativistic quantum mechanics},
  journal = {Reviews of Modern Physics},
  volume  = {20},
  number  = {2},
  pages   = {367--387},
  year    = {1948}
}

@book{oksendal2003,
  author    = {{\O}ksendal, Bernt},
  title     = {Stochastic Differential Equations: An Introduction with Applications},
  edition   = {6},
  publisher = {Springer-Verlag, Berlin},
  year      = {2003}
}

@book{karatzas1991,
  author    = {Karatzas, Ioannis and Shreve, Steven E.},
  title     = {Brownian Motion and Stochastic Calculus},
  edition   = {2},
  series    = {Graduate Texts in Mathematics},
  volume    = {113},
  publisher = {Springer-Verlag, New York},
  year      = {1991}
}

@book{simon1979,
  author    = {Simon, Barry},
  title     = {Functional Integration and Quantum Physics},
  series    = {Pure and Applied Mathematics},
  volume    = {86},
  publisher = {Academic Press, New York},
  year      = {1979}
}

Prerequisites

02.13.03
02.07.04

Tier anchors

beginner: Øksendal Stochastic Differential Equations (6th ed.) Ch. 8 (informal Feynman-Kac picture); Evans An Introduction to Stochastic Differential Equations Ch. 6 (heat equation as an average over paths)
intermediate: Øksendal SDE Ch. 8 (Thm 8.2.1, the Feynman-Kac formula); Karatzas-Shreve Brownian Motion and Stochastic Calculus §4.4 (the Cauchy problem and the killed diffusion)
master: Karatzas-Shreve Brownian Motion and Stochastic Calculus §§4.4, 5.7; Kac On distributions of certain Wiener functionals (1949); Simon Functional Integration and Quantum Physics Ch. 1-3 (Euclidean Schrödinger semigroups)

References

Kac — On distributions of certain Wiener functionals (Trans. Amer. Math. Soc. 65, 1949) · pp. 1--13; the occupation-time formula and the exponential-functional identity now called Feynman-Kac
Feynman — Space-time approach to non-relativistic quantum mechanics (Rev. Mod. Phys. 20, 1948) · pp. 367--387; the sum-over-paths representation of the propagator that Kac transposed to real time
Øksendal — Stochastic Differential Equations: An Introduction with Applications (6th ed., Springer, 2003) · Ch. 8, Thm 8.2.1 (Feynman-Kac) and the Kac moment formula; Ch. 9 (Dirichlet problem and exit times)
Karatzas & Shreve — Brownian Motion and Stochastic Calculus (2nd ed., Springer, 1991) · §4.4 (the Cauchy problem, Thm 4.4.2) and §5.7 (Feynman-Kac, killed diffusions, the Dirichlet problem)
Simon — Functional Integration and Quantum Physics (Academic Press, 1979) · Ch. 1--3; the imaginary-time Schrödinger semigroup $e^{-tH}$ and its Wiener-integral kernel

Estimated time

beginner: 18m
intermediate: 45m
master: 85m