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A ''diffusion process'' is a [[Markov process]] with [[Sample-continuous_process|continuous sample paths]] for which the [[Kolmogorov_equations|Kolmogorov forward equation]] is the [[Fokker–Planck equation]].<ref>{{cite web|title=9. Diffusion processes|url=http://math.nyu.edu/faculty/varadhan/stochastic.fall08/sec10.pdf|access-date=October 10, 2011}}</ref>
A diffusion process is defined by the following properties. Let <math>a^{ij}(x,t)</math> be uniformly continuous coefficients and <math>b^{i}(x,t)</math> be bounded, Borel measurable drift terms. There is a unique family of probability measures <math>\mathbb{P}^{\xi,\tau}_{a;b}</math> (for <math>\tau \ge 0</math>, <math>\xi \in \mathbb{R}^d</math>) on the canonical space <math>\Omega = C([0,\infty), \mathbb{R}^d)</math>, with its Borel <math>\sigma</math>-algebra, such that:▼
▲Let <math>a^{ij}(x,t)</math> be uniformly continuous coefficients and <math>b^{i}(x,t)</math> be bounded, Borel measurable drift terms. There is a unique family of probability measures <math>\mathbb{P}^{\xi,\tau}_{a;b}</math> (for <math>\tau \ge 0</math>, <math>\xi \in \mathbb{R}^d</math>) on the canonical space <math>\Omega = C([0,\infty), \mathbb{R}^d)</math>, with its Borel <math>\sigma</math>-algebra, such that:
1. (Initial Condition) The process starts at <math>\xi</math> at time <math>\tau</math>: <math>\mathbb{P}^{\xi,\tau}_{a;b}[\psi \in \Omega : \psi(t) = \xi \text{ for } 0 \le t \le \tau] = 1.</math>
2. (Local Martingale Property) For every <math>f \in C^{2,1}(\mathbb{R}^d \times [\tau,\infty))</math>, the process
<math>M_t^{[f]} = f(\psi(t),t) - f(\psi(\tau),\tau) - \int_\tau^t \bigl(L_{a;b} + \tfrac{\partial}{\partial s}\bigr) f(\psi(s),s)\,ds</math> is a local martingale under <math>\mathbb{P}^{\xi,\tau}_{a;b}</math> for <math>t \ge \tau</math>, with <math>M_t^{[f]} = 0</math> for <math>t \le \tau</math>.
This family <math>\mathbb{P}^{\xi,\tau}_{a;b}</math> is called the <math>\mathcal{L}_{a;b}</math>-diffusion.
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[[Category:Markov processes]]
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