Stochastic process

A stochastic process is a family of random variables indexed by time or another ordered parameter.

A stochastic process in which individuals independently produce offspring according to specified distributions. It models growing populations and gives an approximation to sparse percolation clusters when collisions between branches can be neglected. On a lattice, loops and overlapping branches prevent this approximation from being an exact identity.

The raw same-observable time correlation $C(t,\tau )=⟨X(t)X(t+\tau )⟩$. For a stationary process, it equals $⟨X{⟩}^2+\mathrm{Cov}(X(t),X(t+\tau ))$. Its connected part is the autocovariance; unlike the normalized statistical autocorrelation, no division by the variance is imposed. The distinction matters for fluctuations around a nonzero equilibrium point.

Near the stable Adler phase equation point $\alpha =\arcsin (\omega /ϵ)$, linearized noise obeys an Ornstein-Uhlenbeck process with relaxation rate $\kappa =\sqrt{{ϵ}^2-{\omega }^2}$. Its harmonic-model raw correlation is ${\alpha }^2+(T_{\mathrm{eff}}/\kappa )e^{-\kappa |\tau |}$. Nonlinear drift also shifts the local mean at order $T_{meff}$. This describes times after relaxation but before appreciable phase slips; it is not a stationary infinite-time correlation of the unwrapped phase at fixed nonzero noise.

For a stochastic process $X_t$ and a fixed sample outcome $\omega$, its sample path is the function $t\mapsto X_t(\omega )$. Statements about continuous, càdlàg or Hölder continuous function sample paths concern these functions simultaneously in time; they differ from assertions made only at each fixed deterministic time. A probability-one path property can be checked using countable dense times when the appropriate continuity is available.

A discrete-time stochastic process has bounded increments if a single deterministic $C<\infty$ satisfies $|X_{n+1}-X_n|\le C$ almost surely for every $n$. The constant is uniform in time and sample outcome; bounding each increment by a different constant does not suffice. This property applies to differences, so it does not require the process itself to stay bounded.

A discrete-time martingale with a common deterministic bound on all increments either converges finitely or has $\mathrm{liminf}{X}_n=-\infty$ and $\mathrm{limsup}{X}_n=+\infty$, almost surely. Stopping on first crossing below $-A$ gives a lower bound $-A-M$ on the overshoot. Adding $A+M+X_0^{-}$ produces a nonnegative martingale, so the Martingale convergence theorem forces finite convergence on the event of never crossing. Apply the same argument to $-X$.

A continuous locally defined process is a pair $(X,T)$ specified on the stochastic interval $[0,T)$, with a lifetime stopping time $T$ approached by an announcing sequence for a stopping time $T_n$. Each stopped process is an ordinary continuous adapted process. This is the convention used for continuous local differential equations; no value at $T$ is required. Restricting a globally defined process gives examples, but finite lifetimes also allow approach to a domain boundary or explosion.

A local solution is a locally defined stochastic process taking values in a specified open domain $U$ and satisfying the stochastic differential equation on every stopped interval before its lifetime. For $dX=b(X)dt+\sigma (X)dB$, the integral equation holds after each announcing stop, with the requisite local drift and noise integrability. A maximal local solution of a stochastic differential equation cannot be extended while remaining in $U$; a finite lifetime can be a boundary hit rather than divergence to infinity.

For a stopping time $T$, the stochastic interval $[0,T)$ means the subset $\{(\omega ,t):0\le t<T(\omega )\}$ of sample-time space. Its endpoint varies with the sample. A locally defined stochastic process on it has no prescribed value at the endpoint. This notation is distinct from an ordinary deterministic interval.

Processes $X,Y$ on the same probability space and index set are modifications of each other if $\mathbb{P}(X_t=Y_t)=1$ for each fixed index $t$. This differs from indistinguishability of stochastic processes, which requires equality at all indices on a single event of probability one. Equality at a countable dense set upgrades to indistinguishability when both processes have continuous paths.

A càdlàg modification of a stochastic process is a modification of a stochastic process whose paths are càdlàg on one event of probability one. Equality almost surely at every separately fixed time does not imply indistinguishability of stochastic processes when the original process lacks path regularity. A stochastic process starting at zero, with independent increments, stationary increments, and stochastic continuity, has a càdlàg modification and is a Lévy process under the intrinsic definition.

A continuous modification of a stochastic process is a modification of a stochastic process whose paths are continuous outside a single null event. For a process originally given only on a countable dense set, the analogous construction gives a continuous extension agreeing simultaneously at all original indices. The Kolmogorov continuity theorem and dyadic increment chaining are standard ways to obtain it.

A linear fluctuating interface smooths height gradients by diffusion and receives additive Gaussian white noise. With periodic or Neumann boundary conditions, its nonzero Fourier modes are independent Ornstein-Uhlenbeck processes if the noise is diagonal in that basis. The unconstrained uniform height is a Brownian zero mode of a fluctuating interface.

For a nonzero Fourier mode satisfying $d{h}_q=-\alpha q^2{h}_{q}dt+\sigma d{W}_q$ with $\alpha >0$, the explicit Ornstein-Uhlenbeck solution and Itô isometry give the stationary second moment ${\sigma }^2/(2\alpha q^2)$. The prefactor depends on the normalization of the mode noise, while the inverse-square dependence follows from diffusive relaxation.

An unpinned uniform height driven by Gaussian white noise is Brownian motion. Its variance grows as ${\sigma }^{2}t$ and it has no stationary distribution on the real height axis for $\sigma >0$. Indeed, a stationary characteristic function would obey $\hat{\mu }(k)=\hat{\mu }(k)e^{-{\sigma }^2{k}^{2}t/2}$, incompatible with continuity at zero. Quotienting out the uniform height can still leave stationary shape fluctuations.

A path probability is a probability measure, or a density relative to a specified reference measure, on histories of a stochastic process. A formal Onsager--Machlup path probability must be interpreted with its discretization and initial-state conditioning specified; conditioning on both endpoints produces a separately normalized bridge measure.

Langevin dynamics describes a dynamical variable driven by deterministic forces and random noise. Underdamped Langevin dynamics retains inertia, whereas Overdamped Langevin dynamics neglects it. At thermal equilibrium, the fluctuation-dissipation relation for a Langevin particle relates Gaussian white noise strength to drag and temperature.

An interacting particle system is a stochastic process whose local states evolve through random transitions whose rates depend on nearby states.

In the contact process, infected vertices recover at rate $\mu$, while each oriented nearest-neighbour edge transmits infection at rate $\lambda$. The empty configuration is absorbing.

Place recovery marks at each vertex according to independent rate-$\mu$ Poisson processes and infection arrows on each oriented nearest-neighbour edge according to independent rate-$\lambda$ Poisson processes. A vertex is infected at time $t$ exactly when a time-directed path from an initially infected vertex reaches it while following arrows and avoiding recovery marks.

Reachability in the graphical representation gives

Time reversal of the Poisson graphical representation, together with reversal of every infection arrow, gives

The survival probability is the probability that the contact process begun from one infected vertex never reaches its absorbing empty state. Its critical infection rate is ${\lambda }_c(\mu )=\inf \{\lambda :\theta (\lambda ,\mu )>0\}$.

The telegraph process moves at one of two opposite constant velocities and reverses direction at Poisson event times. Its position has finite propagation speed and persistent increments.

If speed $s$ and reversal rate $\lambda$ tend to infinity with $s^2/(2\lambda )=D$ fixed, the integrated telegraph velocity converges to Brownian motion with diffusivity $D$.

A stochastic process has stationary increments when the distribution of $X_{t+s}-X_t$ depends only on $s$.

A stochastic process has independent increments when its increments over pairwise disjoint time intervals are independent random variables.

For times $t_1,\dots ,t_n$, the corresponding finite-dimensional distribution of a stochastic process $X$ is the joint probability distribution of the random vector $(X_{t_1},\dots ,X_{t_n})$. The collection of these distributions records every finite set of coordinates of the process.

Consistent probability distributions on finite collections of coordinates in standard Borel spaces determine a unique probability measure on the product cylinder $\sigma$-algebra. In particular repeated products of a fixed distribution construct an independent and identically distributed sequence.

The covariance function of a second-order stochastic process $(X_t)$ isIt is a positive-semidefinite kernel and determines the finite-dimensional distributions of a centered Gaussian process.

If a stochastic process indexed by a subset of ${\mathbb{R}}^d$ satisfiesfor some $p,\beta >0$, then it has a continuous modification. More precisely, that modification is locally $\gamma$-Hölder continuous for every $\gamma <\beta /p$.

For a function on the dyadic rationals in $[0,1]$, let $A_n$ be its largest absolute adjacent increment on the level-$n$ grid. If $K_{\alpha }<\infty$ for $\alpha >0$, the function extends uniquely to a Hölder continuous function withChoose $n$ with $2^{-n}\le |t-s|<2^{1-n}$. The level-$n$ left approximations differ by at most two grid steps, and each finer approximation adds at most one level increment. Thus the difference is bounded by $2{\sum }_{j\ge n}{A}_j\le 2^{-n\alpha }{K}_{\alpha }$. For a stochastic process with $\Vert {\xi }_t-{\xi }_s{\Vert }_p\le C|t-s{|}^{\beta }$, the bound $\Vert A_n{\Vert }_p\le C{2}^{-n(\beta -1/p)}$ makes this series converge in the Lp norm for $0<\alpha <\beta -1/p$.

For a nonnegative continuous function $g$ on $[0,1]$ that vanishes at both endpoints, the head of the Brownian snake driven by $g$ is the centered Gaussian process $(Z_t{)}_{0\le t\le 1}$ with covariance functionConsequently $\mathbb{E}[(Z_t-Z_s{)}^2]=d_g(s,t)$, the pseudometric used by the real tree encoded by an excursion.

If the driving function $g$ is $\alpha$-Hölder continuous, then its Brownian snake has a modification that is $\gamma$-Hölder continuous for every $\gamma <\alpha /2$. Indeed,and the conclusion follows from the Kolmogorov continuity theorem by taking $p$ arbitrarily large.

Chordal Schramm–Loewner evolution in the complex upper half-plane from $0$ to $\infty$ is the random Loewner chain driven by $U_t=\sqrt{\kappa }{B}_t$, where $B$ is standard Brownian motion and $\kappa \ge 0$.

Chordal SLE at parameter four has driver $2{\beta }_t$ in half-plane-capacity time. The Itô formula makes the imaginary part of $\log (g_t(z)-W_t)$ drift-free, a feature used in the SLE4 coupling with a Gaussian free field. The trace is simple and avoids fixed nonzero boundary points.

Radial SLE grows from a boundary point to an interior target. In the unit disc from $1$ to $0$, its conformal-radius parameterization satisfiesThe target remains in the remaining simply connected component; the time parameter records its decreasing conformal radius.

For radial SLE at parameter two, the normalized slit-domain Poisson kernel at the growing tip is a local martingale. This observable provides the coupling to loop-erasure of planar Brownian motion, oriented from the boundary point to the interior target. Its discrete counterpart is a loop-erased random walk.

Among Loewner chains with capacity $2t$, continuous real driver and initial driver zero, scale invariance and the centered domain Markov property of a chordal Loewner chain characterize Schramm–Loewner evolution with some $\kappa \ge 0$. Driving-function reconstruction for a chordal Loewner chain turns the hull properties into stationary independent increments and Brownian scaling for the driver. The continuous Lévy process classification then gives $U_t=bt+\sqrt{\kappa }{W}_t$, and scaling eliminates $b$. Regularity and parametrization are essential hypotheses of this characterization.

With the upper-half-plane branch, ${\theta }_t\in (0,\pi )$ before the Loewner swallowing time. Write $Z_t=X_t+i{Y}_t$. The Itô formula and the Chordal Loewner equation giveAt parameter four its drift vanishes. At other parameters the drift describes the competition between the deterministic conformal evolution and the Brownian driver.

For the SLE angle process, the bounded Dirichlet problem on the unbounded surviving domain has data $\pi$ on the intrinsic boundary mapped to the left of the Loewner driving function and zero on the boundary mapped to its right. For a simple Loewner trace these are its left and right banks, with the negative and positive real rays respectively. The tip is a discontinuity point of the boundary data.

For Schramm–Loewner evolution with $\kappa =4$ and $Z_t=g_t(z)-2W_t$, the Itô formula for the complex logarithm gives $d\log Z_t=-2Z_t^{-1}d{W}_t$ before swallowing. The drift cancels exactly at this parameter. Its real part is $\log |Z_t|$ and its imaginary part is the SLE4 angle martingale. The real-part quadratic variation is $4\int X_t^2/|Z_t{|}^{4}dt$, which also gives a direct non-swallowing argument.

Every fixed $z\in \mathbb{H}$ has infinite interior-point swallowing time for a Loewner chain under Schramm–Loewner evolution at $\kappa =4$. The centered flow is bounded on any finite horizon, making $\log |Z_t|$ bounded above. The one-sided bound criterion for a martingale clock gives a finite logarithmic limit at any putative finite swallowing time. Thus $|Z_t|$ stays away from zero and the imaginary coordinate remains positive, so the differential equation extends, a contradiction. In particular the trace misses each fixed interior point almost surely.

For a fixed interior point and parameter four, the SLE angle process has zero drift and satisfies ${\theta }_t=\arg z+2{\int }_0^t{Y}_s|Z_s{|}^{-2}d{B}_s$. Its values in $(0,\pi )$ make this continuous local martingale a true bounded martingale. The identity $\log ({{\rm Y}}_t/{{\rm Y}}_0)=-[\theta {]}_t$, together with the Dambis-Dubins-Schwarz theorem, prevents its Loewner conformal radius from vanishing at a finite lifetime. Since the parameter-four trace is simple and does not meet the real boundary at positive times, finite swallowing would require a visit to the point and vanishing radius. Thus its Loewner swallowing time is infinite almost surely. This gives a global continuous bounded martingale, not just one defined before swallowing.

For $\kappa >0$, $\rho >0$, let $J_t=|g_t^{\prime }(z)|$, $S_t=\sin {\theta }_t$ and ${{\rm Y}}_t=Y_t/J_t$. Thenis a positive continuous local martingale before the Loewner swallowing time. Indeed the Itô formula gives $d\log J_t=-2(X_t^2-Y_t^2)|Z_t{|}^{-4}dt$, $d\log {{\rm Y}}_t=-4Y_t^2|Z_t{|}^{-4}dt$, andCombining the exponents cancels the drift of $M_t$, including the quadratic-variation correction, leaving $d{M}_t/M_t=-\rho X_t/(\sqrt{\kappa }|Z_t{|}^2)d{B}_t$.

A compact H-hull is a bounded relatively closed subset $A$ of the complex upper half-plane $\mathbb{H}$ such that $\mathbb{H}\setminus A$ is a simply connected domain.

The mapping-out function is the unique conformal map $g_A:\mathbb{H}\setminus A\to \mathbb{H}$ with hydrodynamic normalization at infinity

If a compact H-hull lies in a disc of radius $r$ centred at $\xi \in \mathbb{R}$, the displayed estimate holds for $|z-\xi |>2r$, with an absolute constant $C$. It refines the Laurent expansion uniformly in the hull and supplies the first-order increment needed to derive the Chordal Loewner equation from the Loewner local growth property.

Let $D=\mathbb{H}\setminus K$ and stop planar Brownian motion from $z\in D$ on reaching height $R$ or leaving $D$. Then $\mathrm{Im}{g}_K(z)={\lim }_{R\to \infty }R{\mathbb{P}}_z(\mathrm{Im}{B}_{{\tau }_R}=R)$. The stopped harmonic height is a bounded martingale. The bounded difference $g_K(z)-z$ makes its top-boundary value differ from $R$ by a uniform constant, while optional stopping of the imaginary coordinate bounds the top exit probability by $\mathrm{Im}z/R$. Killing on the hull is essential.

For a compact H-hull, $v(z)=\mathrm{Im}{g}_K(z)$ is harmonic and satisfies $0<v(z)\le \mathrm{Im}z$. Local boundedness and the interior continuity of $g_K^{-1}$ show that $v$ tends to zero at every finite boundary point. The maximum principle for harmonic functions applied to $v-\mathrm{Im}z$, with an exhaustion controlling infinity through the Laurent expansion, proves the inequality without boundary smoothness.

The inverse of a hydrodynamically normalized mapping-out function has expansion $f(w)=w+O(1/w)$ at infinity. If bounded-domain points $z_n$ had unbounded images $w_n=g(z_n)$, then $z_n=f(w_n)=w_n+O(1/w_n)$ would be unbounded, a contradiction. Thus $g$ is bounded even near finite rough boundary points. Its own expansion at infinity then makes $g(z)-z$ uniformly bounded throughout the domain. Reflection is needed only near infinity, not along the entire hull boundary.

For the compact H-hull $(x,x+ih]$, the square-root branch asymptotic to $z-x$ gives the hydrodynamic normalization at infinity of its mapping-out function of a compact H-hull. Its expansion has coefficient $h^2/2$, the half-plane capacity of a vertical slit.

If a compact H-hull lies in the unit disc, compare its tail harmonic measure with that of the empty hull and filled half-disc. The Poisson kernel for the upper half-plane turns the inclusion of Brownian exit events into $x\le g_K(x)\le x+1/x$ for $x>1$. Reflection gives $x+1/x\le g_K(x)\le x$ for $x<-1$.