Celebratio Mathematica — Varadhan

Diffusion theory

Works connected to Hermann Dinges

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D. W. Stroock and S. R. S. Varadhan: “Diffusion processes and martingales. II,” pp. 67–75 in Martingales (Oberwolfach, May 17–30, 1970). Edited by H. Dinges. Lecture Notes in Mathematics 190. Springer (Berlin), 1971. MR 0359025 incollection

Hermann Dinges	Related
Daniel W. Stroock	Related

D. W. Stroock and S. R. S. Varadhan: “Diffusion processes and martingales. I,” pp. 60–66 in Martingales (Oberwolfach, May 17–30, 1970). Edited by H. Dinges. Lecture Notes in Mathematics 190. Springer (Berlin), 1971. MR 0359024 incollection

Let \( a:[0,\infty)\times\mathbb{R}^d\to S_d \) and \( b:[0,\infty)\times\mathbb{R}^d\to\mathbb{R}^d \) be bounded measurable functions, and form the elliptic operator \[ L_t=\frac{1}{2}\sum_{i,j=1}^d a_{ij}(t,x)\frac{\partial^2}{\partial x_i\partial x_j}+\sum_{i=1}^d b_i(t,x)\frac{\partial}{\partial x_i} .\] Given \( s\geq 0 \) and \( x\in\mathbb{R}^d \), our aim is to prove the existence and uniqueness of a measure \( P_{s,x} \) which bears the same relation to \( L_t \) as the \( d \) dimensional Wiener \( W_{s,x} \) measure, conditioned to start from \( x \) at time \( s \), bears to \( \frac{1}{2}\Delta \) (\( \Delta = \sum_{i=1}^d \partial^2/\partial x_i^2 \)). We have succeeded in this program for the case when \( a \) is continuous and positive-definite valued.

Hermann Dinges	Related
Daniel W. Stroock	Related