mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a reversible diffusion is a specific example of a reversible

stochastic process In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appea ...

. Reversible diffusions have an elegant

characterization Characterization or characterisation is the representation of persons (or other beings or creatures) in narrative and dramatic works. The term character development is sometimes used as a synonym. This representation may include direct methods ...

due to the

Russia Russia (, , ), or the Russian Federation, is a List of transcontinental countries, transcontinental country spanning Eastern Europe and North Asia, Northern Asia. It is the List of countries and dependencies by area, largest country in the ...

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

Andrey Nikolaevich Kolmogorov Andrey Nikolaevich Kolmogorov ( rus, Андре́й Никола́евич Колмого́ров, p=ɐnˈdrʲej nʲɪkɐˈlajɪvʲɪtɕ kəlmɐˈɡorəf, a=Ru-Andrey Nikolaevich Kolmogorov.ogg, 25 April 1903 – 20 October 1987) was a Sovi ...

Kolmogorov's characterization of reversible diffusions

Let ''B'' denote a ''d''-

dimension In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...

al standard

Brownian motion Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas). This pattern of motion typically consists of random fluctuations in a particle's position insi ...

; let ''b'' : R^''d'' → R^''d'' be a

Lipschitz continuous In mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there exis ...

vector field. Let ''X'' : , +∞) × Ω → R^''d'' be an Itō diffusion defined on a probability space">Itō diffusion">, +∞) × Ω → R^''d'' be an Itō diffusion defined on a probability space (Ω, Σ, P) and solving the Itō stochastic differential equation

\mathrm X_ = b(X_) \, \mathrm t + \mathrm B_

with square-integrable initial condition, i.e. ''X''₀ ∈ ''L''²(Ω, Σ, P; R^''d''). Then the following are equivalent: * The process ''X'' is reversible with

stationary distribution Stationary distribution may refer to: * A special distribution for a Markov chain such that if the chain starts with its stationary distribution, the marginal distribution of all states at any time will always be the stationary distribution. Assum ...

''μ'' on R^''d''. * There exists a

scalar potential In mathematical physics, scalar potential, simply stated, describes the situation where the difference in the potential energies of an object in two different positions depends only on the positions, not upon the path taken by the object in trav ...

Φ : R^''d'' → R such that ''b'' = −∇Φ, ''μ'' has Radon–Nikodym derivative

\frac = \exp \left( - 2 \Phi (x) \right)

and

\int_ \exp \left( - 2 \Phi (x) \right) \, \mathrm x = 1.

(Of course, the condition that ''b'' be the negative of the

gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gradi ...

of Φ only determines Φ

up to Two Mathematical object, mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R'' * if ''a'' and ''b'' are related by ''R'', that is, * if ''aRb'' holds, that is, * if the equivalence classes of ''a'' and ''b'' wi ...

an additive constant; this constant may be chosen so that exp(−2Φ(·)) is a

probability density function In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can ...

with integral 1.)

References

* {{cite thesis , last = Voß , first = Jochen , title = Some large deviation results for diffusion processes , year = 2004 , publisher = PhD thesis , location = Universität Kaiserslautern , url = https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1559 (See theorem 1.4) Stochastic differential equations Probability theorems