Differentiable Paths Research Articles

Word equations of paths

Let 9* denote the semigroup, under concatenation, of all rectifiable, piecewise continuously differentiable paths in Rn which are not constant on any subinterval and which do not form a straight line on any subinterval. Let w1 = w1(3vI,... , xt), us = f&&h ,-**, x6) denote two words in the letters x1 ,..., xt . Let 9 = F(a, )...) ak) be a free semigroup on a, ,..., a, and suppose u&1 ,.~~, %C~ E 9 with Wl(U1 ,*.*a 2~) = w2(u1 ,..., 14,). Let fr ,..., fk f 33* and set gi = ui(fi ,...) f%j E 53*, z’=l >*.*, t. Then clearly wr(g, ,..., gt) = w&r ,,.., gt). The main theorem of this paper is that every solution of wr = wa in 9” is obtained in this manner. In the event that t = 2, we are able to prove the same theorem without the assumption of differentiability.

Functional integration and the Onsager-Machlup Lagrangian for continuous Markov processes in Riemannian geometries

The Onsager-Machlup Lagrangian defined by the functional-integral representation of general continuous Markov processes is derived explicitly and concisely. Attention is given both to the mathematics and its physical interpretation. The method is based on (i) the consideration of continuous and differentiable paths connecting fixed end points in the convariant propagator, (ii) the application of a recently developed Fourier-series analysis of these trajectories in locally flat spaces, and (iii) the use of the proper transformations between locally Euclidean and globally Riemannian geometries. The spectral analysis allows for arbitrary paths rather than an a priori straight line even in the short-time propagator and avoids any ad hoc discretization rule. Specialized to globally flat spaces the result agrees with formulas given by Stratonovich, Horsthemke and Bach, Graham, and Dekker. It is demonstrated that a unique covariant path integral is equivalent to a whole class of stochastically equivalent lattice expressions.

Derivation of the Onsager-Machlup lagrangian for general diffusion processes by means of Fourier series

We present a both physically and mathematically elegant, brief and novel derivation of the lagrangian occurring in the functional integral representation of general diffusion processes. The derivation relies on (i) the consideration of continuous and differentiable paths between fixed endpoints, and (ii) a proper transformation invariant representation and (iii) a Fourier series analysis of these arbitrary paths. No discretezation rule is required in order to put the continuous action on a lattice. Earlier results from Stratonovich, Horsthemke and Bach, and Graham will be readily recovered and thus be given a better comprehension.

Approximation of finite-energy random processes by continuous processes (Corresp.)

It is shown that any random process with finite energy can be approximated arbitrarily closely by a mean-square continuous process. We obtain approximants that in addition have continuous sample paths with probability 1. By a straightforward extension of these results, the approximating process can be made to have differentiable sample paths. The approximation can be performed in real time, in the sense that the approximating process constructed can be regarded as the output of a causal linear time-invariant system whose input is the finite energy process that is to be approximated.

On the Variance of the Number of Zeros of a Stationary Gaussian Process

For a real, stationary Gaussian process $X(t)$, it is well known that the mean number of zeros of $X(t)$ in a bounded interval is finite exactly when the covariance function $r(t)$ is twice differentiable. Cramer and Leadbetter have shown that the variance of the number of zeros of $X(t)$ in a bounded interval is finite if $(r(t) - r(0))/t$ is integrable around the origin. We show that this condition is also necessary. Applying this result, we then answer the question raised by several authors regarding the connection, if any, between the existence of the variance and the existence of continuously differentiable sample paths. We exhibit counterexamples in both directions.

Differentiable paths and the continuation of solutions of differential equations

Differentiable paths and the continuation of solutions of differential equations