Brownian transition probabilities form the heat semigroup, and Brownian stopping transforms boundary data into harmonic functions. Ito's formula translates the Laplacian into infinitesimal expected change; Feynman-Kac adds killing and source terms. Probability thereby supplies both representations and uniqueness arguments for elliptic and parabolic PDEs.
The heat kernel
Definition. For bounded measurable \(f\), \[ P_tf(x)=\mathbb E_xf(B_t) =\int_{\mathbb R^d}f(y)p_t(x,y)\,\mathrm{d} y, \] where \[ p_t(x,y) =(2\pi t)^{-d/2} \exp\!\left(-\frac{\|y-x\|^2}{2t}\right). \]
Proposition (Gaussian semigroup). \[ p_{t+s}(x,z) = \int_{\mathbb R^d}p_t(x,y)p_s(y,z)\,\mathrm{d} y, \] and consequently \(P_{t+s}=P_tP_s\).
Proof. The sum of independent \(N_d(0,tI_d)\) and \(N_d(0,sI_d)\) vectors is \(N_d(0,(t+s)I_d)\). Convolution of their densities gives the kernel identity, and Fubini gives the operator identity.
Theorem (heat equation). If \(f\) is bounded and continuous, then for \(t>0\), \[ u(t,x)=P_tf(x) \] is smooth and satisfies \[ \partial_tu=\frac12\Delta u, \qquad u(t,\cdot)\to f \] locally uniformly at continuity points as \(t\downarrow0\).
Proof. Direct differentiation of the Gaussian density gives \[ \partial_tp_t(x,y)=\frac12\Delta_xp_t(x,y). \] For \(t\ge\varepsilon>0\), every derivative is dominated by an integrable polynomial times a Gaussian, so differentiation under the integral is valid. Approximate-identity convergence of \(p_t\) gives the initial condition.
The Brownian generator
Theorem. For \(f\in C_b^2(\mathbb R^d)\) with uniformly continuous second derivatives, \[ \lim_{t\downarrow0} \left\| \frac{P_tf-f}{t}-\frac12\Delta f \right\|_\infty=0. \]
Proof. Taylor's formula gives, uniformly in \(x\), \[ f(x+h) =f(x)+\nabla f(x)^\top h +\frac12h^\top\nabla^2f(x)h+r(x,h), \] with \(r(x,h)=o(\|h\|^2)\) uniformly as \(h\to0\). Put \(h=\sqrt t\,Z\), \(Z\sim N_d(0,I_d)\). The linear term has mean zero and \[ \mathbb E[h^\top\nabla^2f(x)h] =t\,\operatorname{tr}\nabla^2f(x)=t\Delta f(x). \] Split the remainder on \(\{\|h\|\le\delta\}\) and its Gaussian tail to show \(\sup_x\mathbb E|r(x,h)|=o(t)\).
Harmonic measure and the Dirichlet problem
Definition. For an open domain \(D\subset\mathbb R^d\), let \[ \tau_D=\inf\{t\ge0:B_t\notin D\}. \] The exit distribution \[ \omega_D^x(A)=P_x(B_{\tau_D}\in A) \] on \(\partial D\) is harmonic measure.
Theorem (probabilistic Dirichlet representation). Let \(D\) be bounded and regular for Brownian motion, and let \(g\in C(\partial D)\). Then \[ u(x)=\mathbb E_xg(B_{\tau_D}) =\int_{\partial D}g(y)\omega_D^x(\mathrm{d} y) \] is harmonic in \(D\), continuous on \(\overline D\), and equals \(g\) on the boundary. It is the unique function with these properties.
Proof. If \(U\Subset D\) and \(\tau_U\) is its exit time, the strong Markov property gives \[ u(x) =\mathbb E_x[ \mathbb E_{B_{\tau_U}}g(B_{\tau_D})] =\mathbb E_xu(B_{\tau_U}). \] Taking \(U\) to be a ball proves the mean-value property, hence harmonicity. Boundary regularity gives \(B_{\tau_D}\to z\) in probability when \(x\to z\in\partial D\), so continuity of \(g\) yields the boundary value. If \(v\) is another solution, then \(v(B_{t\wedge\tau_D})\) is a bounded martingale by Ito's formula. Optional stopping gives \(v(x)=\mathbb E_xg(B_{\tau_D})=u(x)\).
Corollary (mean-value property). If \(h\) is harmonic on a neighborhood of the closed ball \(\overline B(x,r)\), then \[ h(x) = \mathbb E_xh(B_{\tau_{B(x,r)}}) = \frac1{|\partial B(x,r)|} \int_{\partial B(x,r)}h(y)\,\mathrm{d}S(y). \]
Proof. Optional stopping gives the first equality. Rotational invariance of Brownian motion makes its exit point from a ball centered at \(x\), when started at \(x\), uniform on the sphere, giving the second.
The Poisson problem and occupation time
Theorem. Suppose \(D\) is a bounded regular domain and \(f\) is bounded continuous. Then \[ u(x)=\mathbb E_x\int_0^{\tau_D}f(B_s)\,\mathrm{d} s \] solves, under standard regularity, \[ -\frac12\Delta u=f\quad\text{in }D, \qquad u=0\quad\text{on }\partial D. \]
Proof. If \(u\in C^2(D)\cap C(\overline D)\) solves the PDE, Ito's formula at \(t\wedge\tau_D\) gives \[ \mathbb Eu(B_{t\wedge\tau_D})-u(x) = -\mathbb E\int_0^{t\wedge\tau_D}f(B_s)\,\mathrm{d} s. \] Let \(t\to\infty\); the boundary term tends to zero and dominated convergence gives the representation. Conversely, the strong Markov property gives the dynamic programming identity on every compactly contained subdomain; standard elliptic regularity then identifies the generator equation.
Feynman-Kac formula
Theorem. Let \(V\) be bounded below and sufficiently regular, and let \(f\) be bounded. Define \[ u(t,x) = \mathbb E_x\!\left[ \exp\!\left\{-\int_0^tV(B_s)\,\mathrm{d} s\right\} f(B_t)\right]. \] Then \(u\) solves \[ \partial_tu =\frac12\Delta u-Vu, \qquad u(0,x)=f(x). \]
Proof. Assume first that \(u\) is \(C^{1,2}\). For fixed \(T\), apply Ito's product rule to \[ Y_s= \exp\!\left\{-\int_0^sV(B_r)\,\mathrm{d} r\right\} u(T-s,B_s). \] Its drift is \[ e^{-\int_0^sV} \left\{-\partial_tu+\frac12\Delta u-Vu\right\}(T-s,B_s)\,\mathrm{d} s, \] which vanishes by the PDE. Thus \(Y_s\) is a martingale, and comparing \(s=0\) with \(s=T\) gives the expectation formula. Conversely, the Markov property yields the semigroup with killing; differentiating it at zero identifies the generator \(\frac12\Delta-V\), and parabolic regularity supplies the classical solution under the stated assumptions.
For a general diffusion, the Laplacian is replaced by its generator \(b^\top\nabla+\frac12\operatorname{tr}(a\nabla^2)\). The same martingale calculation is the basic bridge from stochastic differential equations to elliptic and parabolic PDEs.