The Markov Property at Deterministic Times

Past Information and Future Increments

At a fixed time \(s\), Brownian motion’s future position combines two parts: its current value \(B_s\), which is already known from the past, and a fresh increment \(B_t-B_s\). The increment is independent of the past, but \(B_t\) itself is generally not. The Markov property makes this distinction precise and turns conditional expectations of functions of \(B_t\) into a transition-semigroup calculation.

Work on a filtered probability space \((\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{P})\) carrying a standard Brownian motion \(B_t\) relative to the filtration: in particular, for \(0\le s<t\), \(B_t-B_s\) is independent of \(\mathcal{F}_s\) and has distribution \(N(0,t-s)\). The past through time \(s\) is represented by \(\mathcal{F}_s\). Equalities between conditional expectations below are understood almost surely.

Independence of the Future-Increment Process

The defining independent-increment property first gives the desired conclusion for any finite collection of future times. The same conclusion applies to the entire continuous future-increment path on a fixed bounded interval.

Proof. For a finite collection of future times, write the vector of values as sums of increments over the successive disjoint intervals with endpoints \(s,s+u_1,\ldots,s+u_n\). Brownian independent increments imply that this vector is independent of \(\mathcal{F}_s\); the first coordinate at \(u_1=0\), if present, is the constant zero and does not affect independence. In particular, for every Borel set \(D\subseteq\mathbb{R}^n\) and \(A\in\mathcal{F}_s\), the probability that \(A\) occurs and the vector lies in \(D\) is the product of their probabilities. To pass to paths, use that the Borel sigma-field of \(C([0,T])\), with its uniform norm, is generated by evaluation at rational times in \([0,T]\). Events restricting finitely many such evaluations form a pi-system generating that sigma-field. For each fixed \(A\in\mathcal{F}_s\), the events \(C\) satisfying \(\mathbb{P}(A\cap\{X^{(s)}\in C\})=\mathbb{P}(A)\mathbb{P}(X^{(s)}\in C)\) form a lambda-system. The pi-lambda theorem therefore extends the finite-evaluation identity to every Borel path event. This proves independence of the future path from \(\mathcal{F}_s\). \(\square\)

In conditional-distribution language, for every bounded Borel function \(g:\mathbb{R}^n\to\mathbb{R}\), the theorem says \(\mathbb{E}[g(B_{s+u_1}-B_s,\ldots,B_{s+u_n}-B_s)\mid\mathcal{F}_s]\) equals the unconditional expectation of that same function. This is a statement about the future increments. Their joint law can have dependence across different future times.

The Transition Semigroup and the Markov Formula

For \(h>0\), Brownian motion’s transition density from \(x\) to \(y\) in time \(h\) is \(p_h(x,y)=(2\pi h)^{-1/2}\exp(-(y-x)^2/(2h))\). For a bounded Borel function \(f:\mathbb{R}\to\mathbb{R}\), define the transition operators \(P_h\) by integrating against this density. Set \(P_0f=f\).

The integral here is an ordinary Lebesgue integral. It is finite for bounded \(f\). These operators are called a semigroup because applying one transition and then another is the same as applying a single transition for the combined time.

Proof. For \(u,v>0\), let \(Z_1,Z_2\) be independent standard normal random variables. By the definition of \(P\), \(P_u(P_vf)(x)=\mathbb{E}[f(x+\sqrt{u}Z_1+\sqrt{v}Z_2)]\). The sum \(\sqrt{u}Z_1+\sqrt{v}Z_2\) is normal with mean zero and variance \(u+v\), so this expectation is \(P_{u+v}f(x)\). If \(u=0\) or \(v=0\), the identity follows from \(P_0f=f\). For the conditional formula, put \(h=t-s\). When \(h>0\), \(B_t=B_s+(B_t-B_s)\), and \(B_t-B_s\) is independent of \(\mathcal{F}_s\) with distribution \(N(0,h)\). Thus the conditional expectation of \(f(B_s+(B_t-B_s))\) is the Gaussian average \(P_hf(B_s)\). This conditional-expectation statement can be verified by integrating both sides over each event \(A\in\mathcal{F}_s\), using independence of the increment and the definition of its normal distribution. When \(h=0\), \(B_t=B_s\) and \(P_0f(B_s)=f(B_s)\), so the identity holds as well. \(\square\)

The theorem also holds for nonnegative Borel \(f\), with possibly infinite expectations, by applying the bounded result to \(f\wedge n\) and taking increasing limits. For a signed Borel function \(f\), it holds whenever \(\mathbb{E}|f(B_t)|<\infty\), by applying the result to its positive and negative parts. The bounded version suffices for the examples below.

A Common Mistake: Confusing Position with Increment

A tempting but incorrect calculation is \(\mathbb{E}[B_t\mid\mathcal{F}_s]=0\), on the grounds that the Brownian motion has mean zero. That uses the unconditional mean of \(B_t\) where a conditional mean is required. For \(s\le t\), the correct decomposition is \(B_t=B_s+(B_t-B_s)\): the first term is known at time \(s\), and the second has conditional mean zero. Hence \(\mathbb{E}[B_t\mid\mathcal{F}_s]=B_s\). For example, at \(s=1,t=2\), the incorrect answer \(0\) fails because taking expectations conditionally must preserve the known value \(B_1\); the corrected answer \(B_1\) does so.

The Markov formula concerns the conditional expectation of a function of the future position, given the present information. It does not say that future positions are independent of the past. The transition semigroup accounts for the current position \(B_s\); the fresh randomness enters only through the increment over the time interval \(t-s\).