Brownian martingales encode its driftless increments in algebraic forms adapted to different questions. Polynomial martingales measure moments, exponential martingales control barriers and change measure, and space-time harmonic functions connect conditional expectation to the heat equation.
Linear and quadratic martingales
Theorem. With respect to the usual Brownian filtration, \[ B_t,\qquad B_t^2-t \] are martingales.
Proof. For \(s<t\), write \(B_t=B_s+Z\), where \(Z\sim N(0,t-s)\) is independent of \(\mathcal F_s\). Hence \[ \mathbb E[B_t\mid\mathcal F_s]=B_s. \] Also \[ \mathbb E[B_t^2\mid\mathcal F_s] =B_s^2+2B_s\mathbb EZ+\mathbb EZ^2 =B_s^2+t-s, \] which gives the compensated-square identity.
Proposition. The processes \[ B_t^3-3tB_t \] and \[ B_t^4-6tB_t^2+3t^2 \] are martingales.
Proof. Expand \((B_s+Z)^3\) and \((B_s+Z)^4\) conditionally on \(\mathcal F_s\), using \[ \mathbb EZ=0,\quad \mathbb EZ^2=t-s,\quad \mathbb EZ^3=0,\quad \mathbb EZ^4=3(t-s)^2. \] The compensating time polynomials cancel the conditional drift.
Exponential martingales
Theorem. For every \(\theta\in\mathbb R\), \[ Z_t^\theta = \exp\!\left\{\theta B_t-\frac12\theta^2t\right\} \] is a positive martingale with mean one.
Proof. For \(s<t\), \[ Z_t^\theta = Z_s^\theta \exp\!\left\{ \theta(B_t-B_s)-\frac12\theta^2(t-s)\right\}. \] The second factor is independent of \(\mathcal F_s\) and has expectation one by the Gaussian moment-generating function. Therefore \(\mathbb E[Z_t^\theta\mid\mathcal F_s]=Z_s^\theta\).
Corollary (maximal tail bound). For \(a>0\), \(T>0\), and \(\theta>0\), \[ P\!\left(\sup_{0\le t\le T} \{\theta B_t-\tfrac12\theta^2t\}\ge a\right) \le e^{-a}. \]
Proof. Apply Doob's weak maximal inequality to the nonnegative martingale \(Z_t^\theta\): \[ P(\sup_{t\le T}Z_t^\theta\ge e^a) \le e^{-a}\mathbb EZ_T^\theta=e^{-a}. \]
Optimizing \(\theta\) turns this inequality into sub-Gaussian boundary bounds. The same exponential martingale is the density process in the Cameron-Martin-Girsanov theorem.
Heat polynomials
Theorem. Define polynomials \(H_n(x,t)\) by the generating function \[ \exp\!\left\{\theta x-\frac12\theta^2t\right\} = \sum_{n=0}^{\infty}\frac{\theta^n}{n!}H_n(x,t). \] Then \(H_n(B_t,t)\) is a martingale for every \(n\). The first terms are \[ H_0=1,\quad H_1=x,\quad H_2=x^2-t,\quad H_3=x^3-3tx. \]
Proof. The exponential generating function is a martingale for every \(\theta\). On compact \(\theta\)-sets its power series can be differentiated term by term in \(L^1\), because Gaussian variables have all exponential moments. Comparing coefficients in \[ \mathbb E[Z_t^\theta\mid\mathcal F_s]=Z_s^\theta \] gives \(\mathbb E[H_n(B_t,t)\mid\mathcal F_s]=H_n(B_s,s)\).
Space-time harmonic functions
Theorem. Let \(u\in C^{1,2}([0,T]\times\mathbb R)\) satisfy the backward heat equation \[ \partial_tu+\frac12\partial_{xx}u=0. \] Under sufficient growth control for integrability, \[ u(t,B_t) \] is a martingale on \([0,T]\).
Proof. Ito's formula gives \[ \mathrm{d} u(t,B_t) = \left(\partial_tu+\frac12\partial_{xx}u\right)(t,B_t)\,\mathrm{d} t +\partial_xu(t,B_t)\,\mathrm{d}B_t. \] The PDE cancels the finite-variation drift. The remaining stochastic integral is a local martingale and is a true martingale under the stated integrability control.
Corollary. If \(f\) is bounded measurable and \[ u(t,x)=P_{T-t}f(x)=\mathbb E[f(x+B_{T-t})], \] then \[ u(t,B_t)=\mathbb E[f(B_T)\mid\mathcal F_t]. \]
Proof. Independent increments give \(B_T=B_t+(B_T-B_t)\), with the second term independent of \(\mathcal F_t\) and distributed as \(B_{T-t}\). Thus the conditional expectation equals \(P_{T-t}f(B_t)\). Smooth approximation gives the heat-equation interpretation when \(f\) is not smooth.
Harmonic functions in several dimensions
Proposition. If \(B_t\) is \(d\)-dimensional Brownian motion and \(h\in C^2(D)\) is harmonic, \(\Delta h=0\), then \[ h(B_{t\wedge\tau_U}) \] is a local martingale for every relatively compact \(U\Subset D\), where \(\tau_U\) is the exit time from \(U\).
Proof. Multidimensional Ito gives \[ h(B_{t\wedge\tau_U})-h(B_0) = \int_0^{t\wedge\tau_U}\nabla h(B_s)^\top\mathrm{d}B_s +\frac12\int_0^{t\wedge\tau_U}\Delta h(B_s)\,\mathrm{d} s. \] The second term vanishes. Localization inside \(U\) controls the stochastic integral.