2016/05/26 by José Villa‐Morales, José Villa-Morales, Villa-Morales, José
Economics, Econometrics and Finance · Mathematics · #60G42 #60J75 #Boundary value problem #Bounded function #Brownian motion #Combinatorics #Dirichlet boundary condition #Dirichlet distribution #Dirichlet problem #FOS: Mathematics #Mathematical Approximation and Integration #Mathematical analysis #Mathematics #Nabla symbol #Omega #Operator (biology) #Physics #Primary 35K20 #Probability (math.PR) #Quantum mechanics #Secondary 60J65 #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #Uniqueness #math.PR #msc:35K20 #msc:60G42 #msc:60J65 #msc:60J75
paper · pdf · doi:10.48550/arxiv.1605.08453
17 pages
arxiv created 2016/05/26 · openalex publication_date 2016/05/26 · arxiv updated 2016/05/30 · openalex created_date 2022/10/01 · openalex updated_date 2026/08/04
In this paper we study the Dirichlet problem corresponding to an open bounded set D⊂ ℝd and the operator A=∑i=1da\frac∂ 2∂ xi2 +∑i=1dbi\frac∂ ∂ xi, where a>0 and b∈ ℝd. We define a mean value property and prove that a function u has such property in D if and only if Au=0 in D. Using this characterization, and a drifted Brownian motion, we define a family of random variables that converges almost surely and the limit is used to give an explicit representation for the solutions to the Dirichlet problem, this immediately implies the uniqueness. On the other hand, the existence of the solution is proved imposing a regular condition on the boundary of D.